3.1.1
3.1.1.1 problem number 150
Added June 9, 2019
Solve the heat equation
\[ u_t = u_{xx} \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{k>0,L>0}], 60*10]];
sol = sol /. { K [1] -> n}
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}\frac {2 e^{-\frac {k n^2 \pi ^2 t}{L^2}} \left (\int _0^L f(x) \sin \left (\frac {n \pi x}{L}\right ) \, dx\right ) \sin \left (\frac {n \pi x}{L}\right )}{L}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t) = k* diff ( u ( x , t ), x, x);
bc := u(0,t) = 0, u(L,t) = 0;
ic := u(x,0) = f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc],u(x,t)) assuming k>0,L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}f \left (x \right ) \sin \left (\frac {n \pi x}{L}\right )d x}{L}\right )\]
Hand
solution
Solve
\[ u_{t}=ku_{xx}\]
BC
\begin{align*} u\left ( 0,t\right ) & =0\qquad t>0\\ u\left ( L,t\right ) & =0\qquad t>0 \end{align*}
Initial conditions
\[ u\left ( x,0\right ) =f\left ( x\right ) \qquad 0<x<L \]
Solution
Using separation of variables, let \(u\left ( x,t\right ) =X\left ( x\right ) T\left ( t\right ) \) . Substituting this back into the PDE gives
\begin{align*} T^{\prime }X & =kX^{\prime \prime }T\\ \frac {1}{k}\frac {T^{\prime }}{T} & =\frac {X^{\prime \prime }}{X}=-\lambda \end{align*}
Where the separation constant is some real value \(-\lambda \) . This gives the following two ODE’s to solve
\begin{align} T^{\prime }+\lambda kT & =0\tag {1}\\ X^{\prime \prime }+\lambda X & =0 \tag {2}\end{align}
Starting with the spatial ODE in order to obtain the eigenvalues and eigenfunctions. The boundary conditions on the spatial ODE become
\begin{align*} X\left ( 0\right ) & =0\\ X\left ( L\right ) & =0 \end{align*}
There are three cases to consider, depending on if \(\lambda <0,\lambda =0,\lambda >0\) . Each one of these cases give a different solution.
Case 1 Assuming \(\lambda <0\) . Therefore \(-\lambda \) is positive say \(\mu \) . The solution (2) can now be written as
\begin{equation} X\left ( x\right ) =c_{1}\cosh \left ( \sqrt {\mu }x\right ) +c_{2}\sinh \left ( \sqrt {\mu }x\right ) \tag {3B}\end{equation}
\[ 0=c_{1}\]
The solution (3B) now reduces to
\[ X\left ( x\right ) =c_{2}\sinh \left ( \sqrt {\mu }x\right ) \]
Applying right side boundary conditions to the above results in
\[ 0=c_{2}\sinh \left ( \sqrt {\mu }L\right ) \]
But \(\sinh \left ( \sqrt {\mu }L\right ) \neq 0\) since it was assumed \(\mu \) is not zero and \(\sinh \) is only zero when its argument is zero. The only possibility then is that \(c_{2}=0,\) which leads to trivial solution. Therefore \(\lambda <0\) is not an
eigenvalue
Case 2 . Assuming \(\lambda =0\) . The ODE becomes \(X^{\prime \prime }=0\) , which has the solution
\[ X\left ( x\right ) =c_{1}x+c_{2}\]
Applying left side B.C. gives
\[ 0=c_{2}\]
The solution now reduces to
\[ X\left ( x\right ) =c_{1}x \]
Applying right side B.C. gives
\[ 0=c_{1}L \]
Leading to the trivial solution. Therefore \(\lambda =0\) is not an eigenvalue
Case 3 Assuming \(\lambda >0\) . In this case equation \(\sqrt {-\lambda }\) is complex and solution to equation (2) can be expressed in terms of trig functions using Euler relation which results in
\begin{equation} X\left ( x\right ) =c_{1}\cos \left ( \sqrt {\lambda }x\right ) +c_{2}\sin \left ( \sqrt {\lambda }x\right ) \tag {4}\end{equation}
\[ 0=c_{1}\]
Solution (4) now reduces to
\begin{equation} X\left ( x\right ) =C_{2}\sin \left ( \sqrt {\lambda }x\right ) \tag {5}\end{equation}
\[ 0=C_{2}\sin \left ( \sqrt {\lambda }L\right ) \]
Non-trivial solution implies \(\sin \left ( \sqrt {\lambda }L\right ) =0\) or \(\sqrt {\lambda }L=n\pi \) for \(n=1,2,3,\cdots \) . Therefore the eigenvalues are
\[ \lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \]
And the corresponding eigenfunctions from (5) are
\begin{equation} X_{n}\left ( x\right ) =C_{n}\sin \left ( \sqrt {\lambda _{n}}x\right ) \tag {6}\end{equation}
\[ T^{\prime }+\lambda _{n}kT=0 \]
This has the solution (using an integrating factor method)
\begin{equation} T_{n}\left ( t\right ) =e^{-\lambda _{n}kt} \tag {7}\end{equation}
fundamental solution is
\[ u_{n}\left ( x,t\right ) =T_{n}\left ( t\right ) X_{n}\left ( x\right ) \]
Linear combination of fundamental solutions is also a solution (since this is a
linear PDE). Therefore the general solution is given by
\begin{align} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }u_{n}\nonumber \\ & =\sum _{n=1}^{\infty }T_{n}\left ( t\right ) X_{n}\left ( x\right ) \nonumber \\ & =\sum _{n=1}^{\infty }C_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \tag {8}\end{align}
The initial condition is now used to determine \(c_{n}\) , . At \(t=0\) , \(u\left ( x,0\right ) =f\left ( x\right ) \) and the above becomes
\[ f\left ( x\right ) =\sum _{n=1}^{\infty }C_{n}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Multiplying both sides of the above equation by eigenfunction \(\sin \left ( \sqrt {\lambda _{m}}x\right ) \) and integrating over the domain of \(f\left ( x\right ) \) gives
\[ \int _{0}^{L}f\left ( x\right ) \sin \left ( \sqrt {\lambda _{m}}x\right ) dx=\int _{0}^{L}\sum _{n=1}^{\infty }C_{n}\sin \left ( \sqrt {\lambda _{n}}x\right ) \sin \left ( \sqrt {\lambda _{m}}x\right ) dx \]
Interchanging the order of summation and integration gives
\[ \int _{0}^{L}f\left ( x\right ) \sin \left ( \sqrt {\lambda _{m}}x\right ) dx=\sum _{n=1}^{\infty }C_{n}\int _{0}^{L}\sin \left ( \sqrt {\lambda _{n}}x\right ) \sin \left ( \sqrt {\lambda _{m}}x\right ) dx \]
By the orthogonality of the sine functions, all terms in the right side vanish except when \(n=m\) , leading to
\begin{align*} \int _{0}^{L}f\left ( x\right ) \sin \left ( \sqrt {\lambda _{m}}x\right ) dx & =C_{m}\int _{0}^{L}\sin ^{2}\left ( \sqrt {\lambda _{m}}x\right ) dx\\ & =C_{m}\frac {L}{2}\end{align*}
Therefore (replacing \(m\) back to \(n\) now, since it is arbitrary)
\[ C_{n}=\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) dx\qquad n=1,2,3,\cdots \]
Summary of solution
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }C_{n}e^{-\left ( \frac {n\pi }{L}\right ) ^{2}\lambda _{n}t}\sin \left ( \frac {n\pi }{L}x\right ) \]
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3.1.1.2 problem number 151
Added June 20, 2019
Solve the heat equation
\[ u_t = k u_{xx} \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition is \(u(x,0)=f(x)\) using the following specific values
\begin{align*} f(x) &= 100\\ k&=\frac {1}{100}\\ L&=10 \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; k =1/100; L=10;
f=100;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}], 60*10]];
sol = sol /. { K [1] -> n}
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}-\frac {200 \left (-1+(-1)^n\right ) e^{-\frac {n^2 \pi ^2 t}{10000}} \sin \left (\frac {n \pi x}{10}\right )}{n \pi }\right \}\right \}\]
Maple ✓
restart ; k :=1/100; L :=10; f :=100; pde := diff ( u ( x , t ), t) = k* diff ( u ( x , t ), x, x);
bc := u(0,t) = 0, u(L,t) = 0;
ic := u(x,0) = f;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc],u(x,t))),output= ' realtime ' ));
\[u \left (x , t\right ) = -\frac {200 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (\left (-1\right )^{n}-1\right ) \sin \left (\frac {n \pi x}{10}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{10000}}}{n}\right )}{\pi }\]
Hand
solution
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }C_{n}e^{-\left ( \frac {n\pi }{L}\right ) ^{2}\lambda _{n}t}\sin \left ( \frac {n\pi }{L}x\right ) \\ C_{n} & =\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\qquad n=1,2,3,\cdots \end{align*}
In this case \(L=10,k=\frac {1}{100},f\left ( x\right ) =100\) . Hence the solution becomes
\begin{align*} C_{n} & =\frac {2}{10}\int _{0}^{10}100\sin \left ( \frac {n\pi }{10}x\right ) dx\\ & =\frac {200}{\pi n}\left ( 1-\cos \left ( n\pi \right ) \right ) \\ & =\frac {200}{\pi n}\left ( 1+\left ( -1\right ) ^{n+1}\right ) \end{align*}
Hence
\[ u\left ( x,t\right ) =\frac {200}{\pi }\sum _{n=1}^{\infty }\frac {1}{n}\left ( 1+\left ( -1\right ) ^{n+1}\right ) e^{-\left ( \frac {n\pi }{10}\right ) ^{2}\lambda _{n}t}\sin \left ( \frac {n\pi }{10}x\right ) \]
The following is an animation of the solution
Source code used for the above
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3.1.1.3 \(u=x(1-x)\) problem number 152
Added June 11, 2019
Solve the heat equation
\[ u_t = \frac {1}{100} u_{xx} \]
For \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition is \(u(x,0)=x(1-x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == 1/100* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == x*(1-x);
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{k>0,L>0}], 60*10]];
sol = sol /. { K [1] -> n}
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}-\frac {4 \left (-1+(-1)^n\right ) e^{-\frac {1}{100} n^2 \pi ^2 t} \sin (n \pi x)}{n^3 \pi ^3}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t) = 1/100* diff ( u ( x , t ), x, x);
bc := u(0,t) = 0, u(1,t) = 0;
ic := u(x,0) = x*(1-x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc],u(x,t))),output= ' realtime ' ));
\[u \left (x , t\right ) = -\frac {4 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (\left (-1\right )^{n}-1\right ) \sin \left (n \pi x \right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{100}}}{n^{3}}\right )}{\pi ^{3}}\]
Hand
solution
Solve \(u_{t}=\frac {1}{100}u_{xx}\) with \(0<x<1\) and initial conditions \(u\left ( x,0\right ) =x\left ( 1-x\right ) \)
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions. Here \(L=1\) and \(k=\frac {1}{100}\) . Hence the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\left ( n\pi \right ) ^{2}t}\sin \left ( n\pi x\right ) \]
At \(t=0\)
\[ x\left ( 1-x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( n\pi x\right ) \]
Multiplying both sides by \(\sin \left ( m\pi x\right ) \) and integrating gives
\[ \int _{0}^{1}x\left ( 1-x\right ) \sin \left ( m\pi x\right ) dx=\int _{0}^{1}\sum _{n=1}^{\infty }B_{n}\sin \left ( n\pi x\right ) \sin \left ( m\pi x\right ) dx \]
By orthogonality of sin function the above simplifies to
\begin{align*} \int _{0}^{1}x\left ( 1-x\right ) \sin \left ( n\pi x\right ) dx & =B_{n}\int _{0}^{1}\sin ^{2}\left ( n\pi x\right ) dx\\ \int _{0}^{1}x\left ( 1-x\right ) \sin \left ( n\pi x\right ) dx & =\frac {1}{2}B_{n}\end{align*}
But \(\int _{0}^{1}x\left ( 1-x\right ) \sin \left ( n\pi x\right ) dx=\frac {2\left ( 1+\left ( -1\right ) ^{n+1}\right ) }{n^{3}\pi ^{3}}\) . Hence
\[ B_{n}=\frac {4\left ( 1+\left ( -1\right ) ^{n+1}\right ) }{n^{3}\pi ^{3}}\]
And the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {4\left ( 1+\left ( -1\right ) ^{n+1}\right ) }{n^{3}\pi ^{3}}e^{-k\left ( n\pi \right ) ^{2}t}\sin \left ( n\pi x\right ) \]
This is animation of the solution for \(30\) seconds. (Animation will only show in the HTML version)
Source code used for the above
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3.1.1.4 problem number 153
This is problem 2.3.3, part (a) from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ u_t = k u_{xx} \]
Subject to boundary conditions \(u(0,t)=0\) and \(u(L,t)=0\) with the temperature initially \(u(x,0)=6 \sin \left ( \frac {9 \pi x}{L} \right )\)
Mathematica ✓
ClearAll [ " Global ` * " ]; NumericQ [ L ] =. ;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == 6* Sin [(9* Pi *x)/L];
NumericQ [ L ] = True ;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
NumericQ [L] =. ;
\[\left \{\left \{u(x,t)\to 6 e^{-\frac {81 \pi ^2 k t}{L^2}} \sin \left (\frac {9 \pi x}{L}\right )\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := u(0,t)=0,u(L,t)=0;
ic := u(x,0)=6* sin (9* Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming k>0,L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 6 \sin \left (\frac {9 \pi x}{L}\right ) {\mathrm e}^{-\frac {81 k \,\pi ^{2} t}{L^{2}}}\]
Hand
solution
Solve \(u_{t}=ku_{xx}\) with \(0<x<L\) and initial conditions \(u\left ( x,0\right ) =6\sin \left ( \frac {9\pi x}{L}\right ) \) .
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions. At \(t=0\)
\[ 6\sin \left ( \frac {9\pi x}{L}\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \]
For \(n=9\)
\[ 6\sin \left ( \frac {9\pi x}{L}\right ) =B_{9}\sin \left ( \frac {9\pi }{L}x\right ) \]
Hence \(B_{9}=6\) and all other terms are zero. Therefore the solution is
\begin{align*} u\left ( x,t\right ) & =B_{9}\sin \left ( \sqrt {\lambda _{9}}x\right ) e^{-k\lambda _{9}t}\\ & =6\sin \left ( \frac {9\pi }{L}x\right ) e^{-k\left ( \frac {9\pi }{L}\right ) ^{2}t}\\ & =6\sin \left ( \frac {9\pi }{L}x\right ) e^{-k\frac {81\pi ^{2}}{L^{2}}t}\end{align*}
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3.1.1.5 problem number 154
Added Feb 10, 2018.
Solve the heat equation
\[ u_t = u_{xx} \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(40,t) &= 0 \end{align*}
Initial condition is
\(u\left (x,0\right ) =\left \{ \begin {array} [c]{ccc}x & 0 \leq x < 20\\ 40-x & 20 \geq x \leq 40 \end {array} \right . \)
Mathematica ✓
ClearAll [ " Global ` * " ]; f = Piecewise [{{x, Inequality [0, LessEqual , x, Less , 20]}, {40 - x, 20 <= x <= 40}}];
pde = D [u[x, t], t] == D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[40, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t], 60*10]];
sol = sol /. K [1] -> n;
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}\frac {640 e^{-\frac {n^2 \pi ^2 t}{1600}} \cos \left (\frac {n \pi }{4}\right ) \sin ^3\left (\frac {n \pi }{4}\right ) \sin \left (\frac {n \pi x}{40}\right )}{n^2 \pi ^2}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); f := piecewise (0< x and x<20,x,20<x and x<40,(40-x));
pde := diff ( u ( x , t ), t )= diff ( u ( x , t ), x$2 ); ic := u(x,0)=f;
bc := u(0,t)=0,u(40,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \frac {160 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi x}{40}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{1600}} \sin \left (\frac {n \pi }{2}\right )}{n^{2}}\right )}{\pi ^{2}}\]
Hand
solution
Solve \(u_{t}=ku_{xx}\) with \(0<x<40\) and initial conditions \(u\left ( x,0\right ) =f\left ( x\right ) \) as hat function.
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions. Here \(L=40\) and \(k=1\) . Hence the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-\left ( \frac {n\pi }{40}\right ) ^{2}t}\sin \left ( \frac {n\pi }{40}x\right ) \]
At \(t=0\)
\[ f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{40}x\right ) \]
Multiplying both sides by \(\sin \left ( m\pi x\right ) \) and integrating gives
\[ \int _{0}^{40}f\left ( x\right ) \sin \left ( \frac {m\pi }{40}x\right ) dx=\int _{0}^{40}\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{40}x\right ) \sin \left ( \frac {m\pi }{40}x\right ) dx \]
By orthogonality of sin function the above simplifies to
\begin{align*} \int _{0}^{40}f\left ( x\right ) \sin \left ( \frac {n\pi }{40}x\right ) dx & =B_{n}\int _{0}^{40}\sin ^{2}\left ( \frac {n\pi }{40}x\right ) dx\\ \int _{0}^{20}x\sin \left ( \frac {n\pi }{40}x\right ) dx+\int _{20}^{40}\left ( 40-x\right ) \sin \left ( \frac {n\pi }{40}x\right ) dx & =20B_{n}\end{align*}
But \(\int _{0}^{20}x\sin \left ( \frac {n\pi }{40}x\right ) dx+\int _{20}^{40}\left ( 40-x\right ) \sin \left ( \frac {n\pi }{40}x\right ) dx=\frac {3200}{n^{2}\pi ^{2}}\sin \left ( \frac {n\pi }{2}\right ) \) . Hence
\[ B_{n}=\frac {160}{n^{2}\pi ^{2}}\sin \left ( \frac {n\pi }{2}\right ) \]
And the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {160}{n^{2}\pi ^{2}}\sin \left ( \frac {n\pi }{2}\right ) e^{-\left ( \frac {n\pi }{40}\right ) ^{2}t}\sin \left ( \frac {n\pi }{40}x\right ) \]
When \(n\) is even the terms are zero Hence
\[ u\left ( x,t\right ) =\sum _{n=1,3,5,\cdots }^{\infty }\frac {160}{n^{2}\pi ^{2}}\sin \left ( \frac {n\pi }{2}\right ) e^{-\left ( \frac {n\pi }{40}\right ) ^{2}t}\sin \left ( \frac {n\pi }{40}x\right ) \]
Or
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {160}{\left ( 2n-1\right ) ^{2}\pi ^{2}}\sin \left ( \frac {\left ( 2n-1\right ) \pi }{2}\right ) e^{-\left ( \frac {\left ( 2n-1\right ) \pi }{40}\right ) ^{2}t}\sin \left ( \frac {\left ( 2n-1\right ) \pi }{40}x\right ) \]
This is animation of the solution for \(400\) seconds. (Animation will only show in the HTML version)
Source code used for the above
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3.1.1.6 problem number 155
This is problem 2.3.3, part (b) from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ \frac { \partial u}{\partial t} = k \frac { \partial ^2 u}{\partial x^2} \]
Subject to boundary conditions \(u(0,t)=0\) and \(u(L,t)=0\) with the temperature initially \(u(x,0)=3 \sin \frac {\pi x}{L} - \sin \frac {3 \pi x}{L}\)
Mathematica ✓
ClearAll [ " Global ` * " ]; NumericQ [ L ] =. ;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == 3* Sin [( Pi *x)/L] - Sin [(3* Pi *x)/L];
NumericQ [ L ] = True ;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
NumericQ [L] =. ;
\[\left \{\left \{u(x,t)\to e^{-\frac {9 \pi ^2 k t}{L^2}} \sin \left (\frac {\pi x}{L}\right ) \left (3 e^{\frac {8 \pi ^2 k t}{L^2}}-2 \cos \left (\frac {2 \pi x}{L}\right )-1\right )\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); assume ( L >0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := u(0,t)=0,u(L,t)=0;
ic := u(x,0)=3* sin ( Pi * x / L )- sin (3* Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \left (3 \sin \left (\frac {\pi x}{L}\right ) {\mathrm e}^{\frac {8 k \,\pi ^{2} t}{L^{2}}}-\sin \left (\frac {3 \pi x}{L}\right )\right ) {\mathrm e}^{-\frac {9 k \,\pi ^{2} t}{L^{2}}}\]
Hand
solution
Solve \(u_{t}=ku_{xx}\) with \(0<x<L\) and initial conditions \(u\left ( x,0\right ) =3\sin \left ( \frac {\pi x}{L}\right ) -\sin \left ( \frac {3\pi x}{L}\right ) \) .
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where the eigenvalues are \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}\) for \(n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions.
Initial conditions are now applied. Setting \(t=0\) , the above becomes
\[ u\left ( x,0\right ) =3\sin \frac {\pi x}{L}-\sin \frac {3\pi x}{L}=\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \]
As the series is unique, the terms coefficients must match for those shown only, and all other \(B_{n}\) terms vanish. This means that by comparing terms
\[ 3\sin \left ( \frac {\pi x}{L}\right ) -\sin \left ( \frac {3\pi x}{L}\right ) =B_{1}\sin \left ( \frac {\pi x}{L}\right ) +B_{3}\sin \left ( \frac {3\pi }{L}x\right ) \]
Therefore
\begin{align*} B_{1} & =3\\ B_{3} & =-1 \end{align*}
And all other \(B_{n}=0\) . The solution is
\[ u\left ( x,t\right ) =3\sin \left ( \frac {\pi }{L}x\right ) e^{-k\left ( \frac {\pi }{L}\right ) ^{2}t}-\sin \left ( \frac {3\pi }{L}x\right ) e^{-k\left ( \frac {3\pi }{L}\right ) ^{2}t}\]
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3.1.1.7 problem number 156
This is problem 2.3.3, part (c) from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ u_t = k u_{xx} \]
Subject to boundary conditions \(u(0,t)=0\) and \(u(L,t)=0\) with the temperature initially \(u(x,0)=2 \cos \frac {3 \pi x}{L}\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == 2* Cos [(3* Pi *x)/L];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> L > 0], 60*10]];
sol = sol/. K [1]->n;
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}\frac {4 \left (1+(-1)^n\right ) e^{-\frac {k n^2 \pi ^2 t}{L^2}} n \sin \left (\frac {n \pi x}{L}\right )}{\left (n^2-9\right ) \pi }\right \}\right \}\]
but
\(n=3\) should be special case
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := u(0,t)=0,u(L,t)=0;
ic := u(x,0)=2* cos (3* Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {-\frac {16 \sin \left (\frac {2 \pi x}{L}\right ) {\mathrm e}^{-\frac {4 k \,\pi ^{2} t}{L^{2}}}}{5}+4 \left (\moverset {\infty }{\munderset {n =4}{\sum }}\frac {\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} n \left (1+\left (-1\right )^{n}\right )}{n^{2}-9}\right )}{\pi }\]
handled
\(n=3\) case correctly.
Hand
solution
Solve \(u_{t}=ku_{xx}\) with \(0<x<L\) and initial conditions \(u\left ( x,0\right ) =2\cos \left ( \frac {3\pi x}{L}\right ) \) .
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions. Initial conditions are now applied. Setting \(t=0\) , the above becomes
\[ u\left ( x,0\right ) =2\cos \left ( \frac {3\pi }{L}x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \]
Multiplying both sides by \(\sin \left ( \frac {m\pi }{L}x\right ) \) and integrating
\begin{align*} \int _{0}^{L}2\cos \left ( \frac {3\pi }{L}x\right ) \sin \left ( \frac {m\pi }{L}x\right ) dx & =\int _{0}^{L}\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \sin \left ( \frac {m\pi }{L}x\right ) dx\\ & =\sum _{n=1}^{\infty }B_{n}\int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) \sin \left ( \frac {m\pi }{L}x\right ) dx \end{align*}
By orthogonality of \(\sin \) functions the above simplifies to
\begin{align*} \int _{0}^{L}2\cos \left ( \frac {3\pi }{L}x\right ) \sin \left ( \frac {m\pi }{L}x\right ) dx & =B_{m}\int _{0}^{L}\sin ^{2}\left ( \frac {m\pi }{L}x\right ) dx\\ & =B_{m}\frac {L}{2}\\ B_{m} & =\frac {4}{L}\int _{0}^{L}\cos \left ( \frac {3\pi }{L}x\right ) \sin \left ( \frac {m\pi }{L}x\right ) dx \end{align*}
For \(m=3,B_{3}=0\) . For \(m\neq 3\)
\begin{align*} B_{m} & =\frac {4}{L}\left ( \frac {1+\left ( -1\right ) ^{m}}{m^{2}-9}\frac {nL}{\pi }\right ) \\ & =\frac {4n}{\pi }\left ( \frac {1+\left ( -1\right ) ^{m}}{m^{2}-9}\right ) \end{align*}
Hence the solution becomes
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1,n\neq 3}^{\infty }\frac {4n}{\pi }\left ( \frac {1+\left ( -1\right ) ^{n}}{n^{2}-9}\right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\\ & =\frac {4}{\pi }\sum _{n=1,n\neq 3}^{\infty }n\left ( \frac {1+\left ( -1\right ) ^{n}}{n^{2}-9}\right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\end{align*}
When \(n\) is odd, all terms become zero, hence the above can be also be written as
\[ u\left ( x,t\right ) =\frac {8}{\pi }\sum _{n=2,4,6,\cdots }^{\infty }\left ( \frac {n}{n^{2}-9}\right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\]
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3.1.1.8 problem number 157
This is problem 2.3.3, part (d) from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ \frac { \partial u}{\partial t} = k \frac { \partial ^2 u}{\partial x^2} \]
Subject to boundary conditions \(u(0,t)=0\) and \(u(L,t)=0\) with the temperature initially \(u\left ( x,0\right ) =\left \{ \begin {array} [c]{ccc}1 & & 0<x\leq \frac {L}{2}\\ 2 & & \frac {L}{2}<x\leq L \end {array} \right . \)
Mathematica ✗
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == Piecewise [{{1, Inequality [0, Less , x, LessEqual , L/2]}, {2, L/2 < x < L}}];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> L > 0], 60*10]];
sol = sol /. K [1] -> n;
$Aborted
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := u(0,t)=0,u(L,t)=0;
ic := u(x,0)= piecewise (0< x and x<=L/2,1,L/2<x and x<L,2);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {{\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \left (1+\cos \left (\frac {n \pi }{2}\right )-2 \left (-1\right )^{n}\right ) \sin \left (\frac {n \pi x}{L}\right )}{n}\right )}{\pi }\]
Hand
solution
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}x\right ) \) are the eigenfunctions. Initial conditions are now applied. Setting \(t=0\) , the above becomes
\begin{equation} f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \tag {3}\end{equation}
\[ f\left ( x\right ) =\left \{ \begin {array} [c]{ccc}1 & & 0<x\leq \frac {L}{2}\\ 2 & & \frac {L}{2}<x<L \end {array} \right . \]
Multiplying both sides of (3) by \(\sin \left ( \frac {m\pi }{L}x\right ) \) and integrating over the domain gives
\[ \int _{0}^{L}\sin \left ( \frac {m\pi }{L}x\right ) f\left ( x\right ) dx=\int _{0}^{L}\left [ \sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {m\pi }{L}x\right ) \sin \left ( \frac {n\pi }{L}x\right ) \right ] dx \]
Interchanging the order of integration and summation
\[ \int _{0}^{L}\sin \left ( \frac {m\pi }{L}x\right ) f\left ( x\right ) dx=\sum _{n=1}^{\infty }\left [ B_{n}\left ( \int _{0}^{L}\sin \left ( \frac {m\pi }{L}x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) \right ] \]
But \(\int _{0}^{L}\sin \left ( \frac {m\pi }{L}x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx=0\) for \(n\neq m\) , hence only one term survives
\[ \int _{0}^{L}\sin \left ( \frac {m\pi }{L}x\right ) f\left ( x\right ) dx=B_{m}\int _{0}^{L}\sin ^{2}\left ( \frac {m\pi }{L}x\right ) dx \]
Renaming \(m\) back to \(n\) and since \(\int _{0}^{L}\sin ^{2}\left ( \frac {m\pi }{L}x\right ) dx=\frac {L}{2}\) the above
becomes
\begin{align*} \int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx & =\frac {L}{2}B_{n}\\ B_{n} & =\frac {2}{L}\int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx\\ & =\frac {2}{L}\left ( \int _{0}^{\frac {L}{2}}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx+\int _{\frac {L}{2}}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx\right ) \\ & =\frac {2}{L}\left ( \int _{0}^{\frac {L}{2}}\sin \left ( \frac {n\pi }{L}x\right ) dx+2\int _{\frac {L}{2}}^{L}\sin \left ( \frac {n\pi }{L}x\right ) dx\right ) \\ & =\frac {2}{L}\left ( \left . \frac {-\cos \left ( \frac {n\pi }{L}x\right ) }{\frac {n\pi }{L}}\right \vert _{0}^{\frac {L}{2}}+2\left . \frac {-\cos \left ( \frac {n\pi }{L}x\right ) }{\frac {n\pi }{L}}\right \vert _{\frac {L}{2}}^{L}\right ) \\ & =\frac {2}{n\pi }\left ( \left ( -\cos \left ( \frac {n\pi }{L}x\right ) \right ) _{0}^{\frac {L}{2}}+2\left ( -\cos \left ( \frac {n\pi }{L}x\right ) \right ) _{\frac {L}{2}}^{L}\right ) \\ & =\frac {2}{n\pi }\left ( \left [ -\cos \left ( \frac {n\pi }{L}\frac {L}{2}\right ) +\cos \left ( 0\right ) \right ] +2\left [ -\cos \left ( n\pi \right ) +\cos \left ( \frac {n\pi }{2}\right ) \right ] \right ) \\ & =\frac {2}{n\pi }\left ( -\cos \left ( \frac {n\pi }{2}\right ) +1-2\cos \left ( n\pi \right ) +2\cos \left ( \frac {n\pi }{2}\right ) \right ) \\ & =\frac {2}{n\pi }\left ( \cos \left ( \frac {n\pi }{2}\right ) +1-2\cos \left ( n\pi \right ) \right ) \end{align*}
Hence the solution is
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\]
With
\begin{align*} B_{n} & =\frac {2}{n\pi }\left ( \cos \left ( \frac {n\pi }{2}\right ) -2\cos \left ( n\pi \right ) +1\right ) \\ & =\frac {2}{n\pi }\left ( 1-2\left ( -1\right ) ^{n}+\cos \left ( \frac {n\pi }{2}\right ) \right ) \\ & =\frac {2}{n\pi }\left ( 1+2\left ( -1\right ) ^{n+1}+\cos \left ( \frac {n\pi }{2}\right ) \right ) \end{align*}
Therefore
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {2}{n\pi }\left ( 1-2\left ( -1\right ) ^{n}+\cos \left ( \frac {n\pi }{2}\right ) \right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\]
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3.1.1.9 problem number 158
Solve the heat equation
\[ \frac { \partial u}{\partial t} = \frac { \partial ^2 u}{\partial x^2} \]
For \(-1<x<1\) and \(t>0\) . The boundary conditions are zero at both ends. Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], {t, 1}] == D [u[x, t], {x, 2}];
ic = u[x, 0] == f[x];
bc = {u[-1, t] == 0, u[1, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}2 e^{-\frac {1}{4} \pi ^2 t K[1]^2} \sin \left (\frac {1}{2} \pi (x+1) K[1]\right ) \int _{-1}^1\frac {1}{2} f(x) \sin \left (\frac {1}{2} \pi (x+1) K[1]\right )dx\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t ) = diff ( u ( x , t ), x$2 ); ic := u(x,0) = f(x);
bc := u(-1,t)=0, u(1,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic, bc],u(x,t)) assuming t>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \moverset {\infty }{\munderset {n =1}{\sum }}\sin \left (\frac {n \pi \left (x +1\right )}{2}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{4}} \int _{-1}^{1}f \left (x \right ) \sin \left (\frac {n \pi \left (x +1\right )}{2}\right )d x\]
Hand
solution
Solve \(u_{t}=ku_{xx}\) with \(-1<x<1\) and initial conditions \(u\left ( x,0\right ) =f\left ( x\right ) \) and B.C. \(u\left ( -1,t\right ) =u\left ( 1,t\right ) =0\) .
Let \(\xi =x+1\) where \(x=-1\cdots 1\) Therefore \(\xi =0\cdots 2\) . This also implies mapping the initial temperature to \(f\left ( x+a\right ) \) . Now the ODE is solved
\begin{align*} u_{t} & =u_{\xi \xi }\\ u\left ( 0,t\right ) & =0\\ u\left ( 2,t\right ) & =0 \end{align*}
The basic solution for this type of PDE was already given in problem 3.1.1.1 on page 329 as
\[ u\left ( \xi ,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\lambda _{n}t}\sin \left ( \sqrt {\lambda _{n}}\xi \right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \) and \(\sin \left ( \sqrt {\lambda _{n}}\xi \right ) \) are the eigenfunctions. Here \(L=2\) . Hence the solution becomes
\[ u\left ( \xi ,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\xi \right ) \]
Changing back to \(x\ \)
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) \]
At \(t=0\)
\[ f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}e^{-k\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) \]
Multiplying both sides by \(\sin \left ( \frac {m\pi }{2}\left ( x+1\right ) \right ) \) and integrating gives
\[ \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {m\pi }{2}\left ( x+1\right ) \right ) dx=\int _{-1}^{1}\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {m\pi }{2}\left ( x+1\right ) \right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx \]
By orthogonality of sin function the above simplifies to
\begin{align*} \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx & =B_{n}\int _{-1}^{1}\sin ^{2}\left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx\\ \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx & =B_{n}\end{align*}
Hence
\[ B_{n}=\int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx \]
And the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx\right ) e^{-k\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) \]
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3.1.1.10 problem number 159
Solve the heat equation
\[ \frac { \partial u}{\partial t} = \frac { \partial ^2 u}{\partial x^2} \]
For \(-1<x<1\) and \(t>0\) . The boundary conditions are zero at both ends. Initial condition is \(u(x,0)=1-x^2\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], {t, 1}] == D [u[x, t], {x, 2}];
ic = u[x, 0] == 1-x^2;
bc = {u[-1, t] == 0, u[1, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}-\frac {16 \left (-1+(-1)^{K[1]}\right ) e^{-\frac {1}{4} \pi ^2 t K[1]^2} \sin \left (\frac {1}{2} \pi (x+1) K[1]\right )}{\pi ^3 K[1]^3}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t ) = diff ( u ( x , t ), x$2 ); ic := u(x,0) = 1-x^2;
bc := u(-1,t)=0, u(1,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic, bc],u(x,t)) assuming t>0),output= ' realtime ' ));
\[u \left (x , t\right ) = -\frac {16 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi \left (x +1\right )}{2}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{4}} \left (\left (-1\right )^{n}-1\right )}{n^{3}}\right )}{\pi ^{3}}\]
Hand
solution
Solve \(u_{t}=u_{xx}\) with \(-1<x<1\) and initial conditions \(u\left ( x,0\right ) =f\left ( x\right ) =1-x^{2}\) and B.C. \(u\left ( -1,t\right ) =u\left ( 1,t\right ) =0\) .
The basic solution for this type of PDE was already given in problem 3.1.1.9 on page 365 as
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx\right ) e^{-k\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) \]
Replacing \(f\left ( x\right ) =1-x^{2}\) then
\begin{align*} \int _{-1}^{1}f\left ( x\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx & =\int _{-1}^{1}\left ( 1-x^{2}\right ) \sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) dx\\ & =\frac {16\left ( 1+\left ( -1\right ) ^{n+1}\right ) }{n^{3}\pi ^{3}}\end{align*}
Hence the solution becomes, using \(k=1\)
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {16\left ( 1+\left ( -1\right ) ^{n+1}\right ) }{n^{3}\pi ^{3}}e^{-\left ( \frac {n\pi }{2}\right ) ^{2}t}\sin \left ( \frac {n\pi }{2}\left ( x+1\right ) \right ) \]
This is animation of the solution for \(2\) seconds. (Animation will only show in the HTML version)
Source code used for the above
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3.1.1.11 problem number 160
Added July 3, 2019
Solve the heat equation for \(u(x,t)\)
\[ u_t= k u_{xx} + Q(x) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t)&=0 \\ u(L,t) &= 0\\ \end{align*}
Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Q[x];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{k>0,L>0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} \left (\frac {\left (1-e^{-\frac {k \pi ^2 t K[1]^2}{L^2}}\right ) \left (\int _0^L \frac {\sqrt {2} Q(x) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}} \, dx\right ) L^2}{k \pi ^2 K[1]^2}+e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} \int _0^L \frac {\sqrt {2} f(x) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}} \, dx\right ) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k* diff ( u ( x , t ), x$2 ) + Q(x);
bc := u(0, t) = 0, u(L, t) = 0;
ic := u(x, 0) = f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc], u(x, t)) assuming L>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {-\int _{0}^{x}\int _{0}^{\textit {\_a}}Q \left (\textit {\_a} \right )d \textit {\_a} d \textit {\_a} L^{2}+\int _{0}^{L}\int _{0}^{\textit {\_z1}}Q \left (\textit {\_z1} \right )d \textit {\_z1} d \textit {\_z1} x L +2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} k t}{L^{2}}} \int _{0}^{L}\left (\int _{0}^{\textit {\_a}}\int _{0}^{\textit {\_z1}}Q \left (\textit {\_z1} \right )d \textit {\_z1} d \textit {\_z1} L +f \left (\textit {\_a} \right ) k L -\int _{0}^{L}\int _{0}^{\textit {\_z1}}Q \left (\textit {\_z1} \right )d \textit {\_z1} d \textit {\_z1} \textit {\_a} \right ) \sin \left (\frac {n \pi \textit {\_a}}{L}\right )d \textit {\_a} \right )}{k \,L^{2}}\]
Hand
solution
Let the solution be
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \Phi _{n}\left ( x\right ) \tag {1}\end{equation}
\(\Phi _{n}\left ( x\right ) \) are the eigenfunctions of the eigenvalue problem for the pde \(u_{t}=ku_{xx}\) with \(u\left ( 0,x\right ) =0,u\left ( L,x\right ) =0\) . This was solved in problem 3.1.1.1 on page 329 and \(\Phi _{n}\left ( x\right ) \) was found as
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
Hence (1) becomes
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \tag {1A}\end{equation}
\[ \sum _{n=1}^{\infty }a_{n}^{\prime }\left ( t\right ) \Phi _{n}\left ( x\right ) =k\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \Phi _{n}^{\prime \prime }\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\Phi _{n}\left ( x\right ) \]
Where \(Q\left ( x\right ) =\sum _{n=1}^{\infty }q_{n}\Phi _{n}\left ( x\right ) \) . Since \(\Phi _{n}^{\prime \prime }\left ( x\right ) =-\lambda _{n}\Phi _{n}\left ( x\right ) \) , then the above simplifies to
\begin{align} \sum _{n=1}^{\infty }a_{n}^{\prime }\left ( t\right ) \Phi _{n}\left ( x\right ) & =-k\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \lambda _{n}\Phi _{n}\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\Phi _{n}\left ( x\right ) \nonumber \\ a_{n}^{\prime }\left ( t\right ) +k\lambda _{n}a_{n}\left ( t\right ) & =q_{n}\tag {2}\end{align}
We now need to find \(q_{n}\) . By orthogonality
\[ \int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx=\int _{0}^{L}q_{n}\Phi _{n}^{2}\left ( x\right ) dx \]
But \(\int _{0}^{L}\Phi _{n}^{2}\left ( x\right ) dx=\int _{0}^{L}\sin ^{2}\left ( \frac {n\pi }{L}x\right ) dx=\frac {L}{2}\) and the above becomes
\[ q_{n}=\frac {2}{L}\int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx \]
Using the above in (2) gives
\[ a_{n}^{\prime }\left ( t\right ) +k\lambda _{n}a_{n}\left ( t\right ) =\frac {2}{L}\int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx \]
The integrating factor is \(I=e^{k\lambda _{n}t}\) . Hence the above becomes
\begin{align*} \frac {d}{dt}\left ( a_{n}\left ( t\right ) e^{k\lambda _{n}t}\right ) & =\frac {2}{L}e^{k\lambda _{n}t}\int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx\\ a_{n}\left ( t\right ) e^{k\lambda _{n}t} & =\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}s}\left ( \int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx\right ) ds+a_{n}\left ( 0\right ) \\ a_{n}\left ( t\right ) & =a_{n}\left ( 0\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}s}\left ( \int _{0}^{L}Q\left ( x\right ) \Phi _{n}\left ( x\right ) dx\right ) ds \end{align*}
Therefore the solution (1A) becomes
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left [ a_{n}\left ( 0\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ] \Phi _{n}\left ( x\right ) \tag {3}\end{equation}
\(t=0\)
\begin{align*} f\left ( x\right ) & =\sum _{n=1}^{\infty }a_{n}\left ( 0\right ) \Phi _{n}\left ( x\right ) \\ \int _{0}^{L}f\left ( x\right ) \Phi _{n}\left ( x\right ) dx & =\int _{0}^{L}a_{n}\left ( 0\right ) \Phi _{n}^{2}\left ( x\right ) dx\\ a_{n}\left ( 0\right ) & =\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \Phi _{n}\left ( x\right ) dx \end{align*}
The solution (3) becomes
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }\left [ \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ] \Phi _{n}\left ( x\right ) \\ & =\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) +\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ) \end{align*}
Where
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
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3.1.1.12 problem number 161
Added July 4, 2019
Solve the heat equation for \(u(x,t)\)
\[ u_t= k u_{xx} + Q(x) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &=0 \\ u(L,t) &= 0\\ \end{align*}
Initial condition is \(u(x,0)=f(x)\) using the following values
\begin{align*} L &=5 \\ k &= \frac {1}{10}\\ f(x)&=x(5-x)\\ Q(x)&=x \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; L =5; k=1/10; f=x*(5-x); Q=x;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Q;
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{k>0,L>0}], 60*10]];
sol = sol/. K [1]->n;
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}-\frac {100 \left (-1+(-1)^n\right ) e^{-\frac {1}{250} n^2 \pi ^2 t} \sin \left (\frac {n \pi x}{5}\right )}{n^3 \pi ^3}+\underset {n=1}{\overset {\infty }{\sum }}-\frac {2500 (-1)^n \left (1-e^{-\frac {1}{250} n^2 \pi ^2 t}\right ) \sin \left (\frac {n \pi x}{5}\right )}{n^3 \pi ^3}\right \}\right \}\]
Maple ✓
restart ; L =5; k =1/10; f = x *(5- x ); Q = x ; pde := diff ( u ( x , t ), t )= k* diff ( u ( x , t ), x$2 ) + Q;
bc := u(0, t) = 0, u(L, t) = 0;
ic := u(x, 0) = f;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc], u(x, t)) assuming L>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {4 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \left (-\pi ^{2} f k \,n^{2}+L^{2} Q \right ) \left (\left (-1\right )^{n}-1\right )}{n^{3}}\right )+\left (L x -x^{2}\right ) Q \,\pi ^{3}}{2 k \,\pi ^{3}}\]
Hand
solution
Solve
\begin{align*} u_{t} & =ku_{xx}+Q\left ( x\right ) \\ u\left ( 0,t\right ) & =0\\ u\left ( L,0\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
With \(k=\frac {1}{10},L=5,f\left ( x\right ) =x\left ( 5-x\right ) ,Q\left ( x\right ) =x.\)
The general problem above was solved in 3.1.1.11 on page 374 and the solution is
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) +\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ) \]
Where
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
Substituting the specific values given above into this solution gives
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }e^{-\frac {1}{10}\left ( \frac {n\pi }{5}\right ) ^{2}t}\sin \left ( \frac {n\pi }{5}x\right ) \left ( \frac {2}{5}\int _{0}^{5}s\left ( 5-s\right ) \sin \left ( \frac {n\pi }{5}s\right ) ds\right ) \\ & +\sum _{n=1}^{\infty }e^{-\frac {1}{10}\left ( \frac {n\pi }{5}\right ) ^{2}t}\sin \left ( \frac {n\pi }{5}x\right ) \left ( \int _{0}^{t}\frac {2}{5}e^{\frac {1}{10}\left ( \frac {n\pi }{5}\right ) ^{2}\tau }\left ( \int _{0}^{5}s\sin \left ( \frac {n\pi }{5}s\right ) ds\right ) d\tau \right ) \end{align*}
Animation is below
Source code used for the above
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3.1.1.13 problem number 162
Added June 23, 2019
Solve the heat equation for \(u(x,t)\)
\[ u_t= k u_{xx} + Q(x,t) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t)&=0 \\ u(L,t) &= 0\\ \end{align*}
Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Q[x,t];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{k>0,L>0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} \left (\int _0^t e^{-\frac {k \pi ^2 K[1]^2 (t-K[2])}{L^2}} \text {Integrate}\left [\frac {\sqrt {2} Q(x,K[2]) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}},\{x,0,L\},\text {Assumptions}\to k>0\land L>0\right ] \, dK[2]+e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} \int _0^L \frac {\sqrt {2} f(x) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}} \, dx\right ) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k* diff ( u ( x , t ), x$2 ) + Q(x,t);
bc := u(0, t) = 0, u(L, t) = 0;
ic := u(x, 0) = f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc], u(x, t)) assuming L>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}f \left (\tau \right ) \sin \left (\frac {n \pi \tau }{L}\right )d \tau \right )+2 \int _{0}^{t}\left (\moverset {\infty }{\munderset {n =1}{\sum }}\sin \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} \left (t -\tau \right )}{L^{2}}} \int _{0}^{L}Q \left (x , \tau \right ) \sin \left (\frac {n \pi x}{L}\right )d x \right )d \tau }{L}\]
Hand
solution
Solving
\begin{align} u_{t} & =kv_{xx}+Q\left ( x,t\right ) \tag {1}\\ u\left ( 0,t\right ) & =0\nonumber \\ u\left ( L,t\right ) & =0\nonumber \\ u\left ( x,0\right ) & =f\left ( x\right ) \nonumber \end{align}
Let the solution be
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \Phi _{n}\left ( x\right ) \tag {1}\end{equation}
\(\Phi _{n}\left ( x\right ) \) are the eigenfunctions of the eigenvalue problem for the pde \(u_{t}=ku_{xx}\) with \(u\left ( 0,x\right ) =0,u\left ( L,x\right ) =0\) . This was solved in problem 3.1.1.1 on page 329 and \(\Phi _{n}\left ( x\right ) \) was found as
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
Hence (1) becomes
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \tag {1A}\end{equation}
\[ \sum _{n=1}^{\infty }a_{n}^{\prime }\left ( t\right ) \Phi _{n}\left ( x\right ) =k\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \Phi _{n}^{\prime \prime }\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\left ( t\right ) \Phi _{n}\left ( x\right ) \]
Where \(Q\left ( x,t\right ) =\sum _{n=1}^{\infty }q_{n}\left ( t\right ) \Phi _{n}\left ( x\right ) \) . Since \(\Phi _{n}^{\prime \prime }\left ( x\right ) =-\lambda _{n}\Phi _{n}\left ( x\right ) \) , then the above simplifies to
\begin{align} \sum _{n=1}^{\infty }a_{n}^{\prime }\left ( t\right ) \Phi _{n}\left ( x\right ) & =-k\sum _{n=1}^{\infty }a_{n}\left ( t\right ) \lambda _{n}\Phi _{n}\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\left ( t\right ) \Phi _{n}\left ( x\right ) \nonumber \\ a_{n}^{\prime }\left ( t\right ) +k\lambda _{n}a_{n}\left ( t\right ) & =q_{n}\left ( t\right ) \tag {2}\end{align}
We now need to find \(q_{n}\left ( t\right ) \) . By orthogonality
\[ \int _{0}^{L}Q\left ( x,t\right ) \Phi _{n}\left ( x\right ) dx=\int _{0}^{L}q_{n}\left ( t\right ) \Phi _{n}^{2}\left ( x\right ) dx \]
But \(\int _{0}^{L}\Phi _{n}^{2}\left ( x\right ) dx=\int _{0}^{L}\sin ^{2}\left ( \frac {n\pi }{L}x\right ) dx=\frac {L}{2}\) and the above becomes
\[ q_{n}\left ( t\right ) =\frac {2}{L}\int _{0}^{L}Q\left ( x,t\right ) \Phi _{n}\left ( x\right ) dx \]
Using the above in (2) gives
\[ a_{n}^{\prime }\left ( t\right ) +k\lambda _{n}a_{n}\left ( t\right ) =\frac {2}{L}\int _{0}^{L}Q\left ( x,t\right ) \Phi _{n}\left ( x\right ) dx \]
The integrating factor is \(I=e^{k\lambda _{n}t}\) . Hence the above becomes
\begin{align*} \frac {d}{dt}\left ( a_{n}\left ( t\right ) e^{k\lambda _{n}t}\right ) & =\frac {2}{L}e^{k\lambda _{n}t}\int _{0}^{L}Q\left ( x,t\right ) \Phi _{n}\left ( x\right ) dx\\ a_{n}\left ( t\right ) e^{k\lambda _{n}t} & =\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}s}\left ( \int _{0}^{L}Q\left ( x,s\right ) \Phi _{n}\left ( x\right ) dx\right ) ds+a_{n}\left ( 0\right ) \\ a_{n}\left ( t\right ) & =a_{n}\left ( 0\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}s}\left ( \int _{0}^{L}Q\left ( x,s\right ) \Phi _{n}\left ( x\right ) dx\right ) ds \end{align*}
Therefore the solution (1A) becomes
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left [ a_{n}\left ( 0\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s,\tau \right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ] \Phi _{n}\left ( x\right ) \tag {3}\end{equation}
\(t=0\)
\begin{align*} f\left ( x\right ) & =\sum _{n=1}^{\infty }a_{n}\left ( 0\right ) \Phi _{n}\left ( x\right ) \\ \int _{0}^{L}f\left ( x\right ) \Phi _{n}\left ( x\right ) dx & =\int _{0}^{L}a_{n}\left ( 0\right ) \Phi _{n}^{2}\left ( x\right ) dx\\ a_{n}\left ( 0\right ) & =\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \Phi _{n}\left ( x\right ) dx \end{align*}
The solution (3) becomes
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }\left [ \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) e^{-k\lambda _{n}t}+e^{-k\lambda _{n}t}\int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s,\tau \right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ] \Phi _{n}\left ( x\right ) \\ & =\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) +\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s,\tau \right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ) \end{align*}
Where
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
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3.1.1.14 problem number 163
Added July 5, 2019
Solve the heat equation for \(u(x,t)\)
\[ u_t= k u_{xx} + Q(x,t) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t)&=0 \\ u(L,t) &= 0\\ \end{align*}
Initial condition is \(u(x,0)=f(x)\) . Using the following values
\begin{align*} L &=\pi \\ k &= \frac {1}{300}\\ Q(x,t) &= e^{-t} \sin (3 x)\\ f(x) &= x(\pi -x) \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; L = Pi ; f=x*( Pi -x); k=1/300; Q= Exp [-t]* Sin [3*x];
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Q;
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}], 60*10]];
sol = sol /. K [1] -> n
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}-\frac {4 \left (-1+(-1)^n\right ) e^{-\frac {n^2 t}{300}} \sin (n x)}{n^3 \pi }+\frac {100}{97} e^{-t} \left (e^{97 t/100}-1\right ) \sin (3 x)\right \}\right \}\]
Maple ✓
restart ; L := Pi ; f := x *( Pi - x ); k :=1/300; Q := exp (- t )* sin (3* x ); pde := diff ( u ( x , t ), t )= k* diff ( u ( x , t ), x$2 ) + Q;
bc := u(0, t) = 0, u(L, t) = 0;
ic := u(x, 0) = f;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc], u(x, t))),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {-\frac {100 \,{\mathrm e}^{-t} \sin \left (3 x \right ) \pi }{97}+\frac {100 \sin \left (3 x \right ) {\mathrm e}^{-\frac {3 t}{100}} \pi }{97}-4 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (-1+\left (-1\right )^{n}\right ) \sin \left (n x \right ) {\mathrm e}^{-\frac {n^{2} t}{300}}}{n^{3}}\right )}{\pi }\]
Hand
solution
Solving
\begin{align*} u_{t} & =ku_{xx}+Q\left ( x,t\right ) \\ u\left ( 0,t\right ) & =0\\ u\left ( L,0\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
With \(k=\frac {1}{300},L=\pi ,f\left ( x\right ) =x\left ( \pi -x\right ) ,Q\left ( x\right ) =e^{-t}\sin \left ( 3x\right ) .\)
The general problem above was solved in 3.1.6.4 on page 694 and the solution is
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \frac {2}{L}\int _{0}^{L}f\left ( s\right ) \Phi _{n}\left ( s\right ) ds\right ) \\ & +\sum _{n=1}^{\infty }e^{-k\lambda _{n}t}\Phi _{n}\left ( x\right ) \left ( \int _{0}^{t}\frac {2}{L}e^{k\lambda _{n}\tau }\left ( \int _{0}^{L}Q\left ( s,\tau \right ) \Phi _{n}\left ( s\right ) ds\right ) d\tau \right ) \end{align*}
Where
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( \sqrt {\lambda _{n}}x\right ) \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \end{align*}
Substituting the specific values given above into this solution gives
\begin{align*} \Phi _{n}\left ( x\right ) & =\sin \left ( nx\right ) \\ \lambda _{n} & =n^{2}\qquad n=1,2,3,\cdots \end{align*}
And
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) \left ( \frac {2}{\pi }\int _{0}^{\pi }s\left ( \pi -s\right ) \sin \left ( ns\right ) ds\right ) \\ & +\sum _{n=1}^{\infty }e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) \left ( \int _{0}^{t}\frac {2}{\pi }e^{\frac {1}{300}n^{2}\tau }e^{-\tau }\left ( \int _{0}^{\pi }\sin \left ( 3s\right ) \sin \left ( ns\right ) ds\right ) d\tau \right ) \end{align*}
But \(\int _{0}^{\pi }s\left ( \pi -s\right ) \sin \left ( ns\right ) ds=\frac {2-2\left ( -1\right ) ^{n}}{n^{3}}\) and \(\int _{0}^{\pi }\sin \left ( 3s\right ) \sin \left ( ns\right ) ds=0\) when \(n\neq 3\) and for \(n=3\) it is \(\frac {\pi }{2}\) . Hence the above reduces to
\begin{align*} u\left ( x,t\right ) & =4\sum _{n=1}^{\infty }\left ( \frac {1-\left ( -1\right ) ^{n}}{\pi n^{3}}\right ) e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) +e^{-\frac {1}{300}\left ( 3\right ) ^{2}t}\sin \left ( 3x\right ) \left ( \int _{0}^{t}\frac {2}{\pi }e^{\frac {1}{300}\left ( 9\right ) ^{2}\tau }e^{-\tau }\left ( \frac {\pi }{2}\right ) d\tau \right ) \\ & =4\sum _{n=1}^{\infty }\left ( \frac {1-\left ( -1\right ) ^{n}}{\pi n^{3}}\right ) e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) +e^{-\frac {9t}{300}}\sin \left ( 3x\right ) \left ( \int _{0}^{t}e^{\frac {9\tau }{300}}e^{-\tau }d\tau \right ) \end{align*}
But \(\int _{0}^{t}e^{\frac {9\tau }{300}}e^{-\tau }d\tau =\frac {100}{97}-\frac {100e^{\frac {-97}{100}t}}{97}\) . Hence the above becomes
\begin{align*} u\left ( x,t\right ) & =4\sum _{n=1}^{\infty }\left ( \frac {1-\left ( -1\right ) ^{n}}{\pi n^{3}}\right ) e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) +e^{-\frac {9t}{300}}\sin \left ( 3x\right ) \left ( \frac {100}{97}-\frac {100e^{\frac {-97}{100}t}}{97}\right ) \\ & =4\sum _{n=1}^{\infty }\left ( \frac {1-\left ( -1\right ) ^{n}}{\pi n^{3}}\right ) e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) +\sin \left ( 3x\right ) \left ( \frac {100}{97}e^{-\frac {3t}{100}}-\frac {100e^{-t}}{97}\right ) \\ & =4\sum _{n=1}^{\infty }\left ( \frac {1-\left ( -1\right ) ^{n}}{\pi n^{3}}\right ) e^{-\frac {1}{300}n^{2}t}\sin \left ( nx\right ) -\frac {100e^{-t}}{97}\sin \left ( 3x\right ) +\frac {100}{97}e^{-\frac {3t}{100}}\sin \left ( 3x\right ) \end{align*}
Animation is below
Source code used for the above
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3.1.1.15 problem number 164
Added April 3, 2019.
Exam question. Math 4567 UMN. Spring 2019.
Solve the heat equation
\[ u_t = u_{xx} + t(\pi -x) \]
For \(0<x<\pi \) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(\pi ,t) &=0 \end{align*}
Initial condition is \(u(x,0)=0\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == D [u[x, t], {x, 2}] + t*( Pi -x);
bc = {u[0,t] == 0, u[ Pi ,t] == 0};
ic = u[x, 0] == 0;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t, Assumptions -> t > 0], 60*10]];
sol = sol /. K [1] -> n;
\[\left \{\left \{u(x,t)\to \underset {n=1}{\overset {\infty }{\sum }}\frac {2 \left (t n^2+e^{-n^2 t}-1\right ) \sin (n x)}{n^5}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= diff ( u ( x , t ), x$2 )+ t *( Pi - x ); ic := u(x,0)=0;
bc := u(0,t)=0, u( Pi , t )=0; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde,ic,bc],u(x,t)) assuming t>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (n x \right ) \left ({\mathrm e}^{-n^{2} t}+n^{2} t -1\right )}{n^{5}}\right )\]
Hand
solution
Let \(u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\left ( t\right ) X_{n}\left ( x\right ) \) , where \(X_{n}\left ( x\right ) \) are the eigenfunctions of the homogeneous PDE \(u_{t}=u_{xx}\) . These are known to be \(X_{n}\left ( x\right ) =\sin \left ( \sqrt {\lambda _{n}}x\right ) \) where \(\lambda _{n}=n^{2},n=1,2,3,\cdots \) . Hence
\[ X_{n}\left ( x\right ) =\sin \left ( nx\right ) \qquad n=1,2,3,\cdots \]
Therefore the solution becomes
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\left ( t\right ) \sin \left ( nx\right ) \tag {A}\end{equation}
\(u_{t}=u_{xx}+t\left ( \pi -x\right ) \) gives
\[ \sum _{n=1}^{\infty }B_{n}^{\prime }\left ( t\right ) X_{n}\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\left ( t\right ) X_{n}^{\prime \prime }\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\left ( t\right ) X_{n}\left ( x\right ) \]
Where \(\sum _{n=1}^{\infty }q_{n}\left ( t\right ) X_{n}\left ( x\right ) \) is the Fourier series expansion of \(t\left ( \pi -x\right ) \) . But \(X_{n}^{\prime \prime }\left ( x\right ) =-\lambda _{n}X_{n}\left ( x\right ) =-n^{2}X_{n}\left ( x\right ) \) . Hence the above becomes
\begin{align} \sum _{n=1}^{\infty }B_{n}^{\prime }\left ( t\right ) X_{n}\left ( x\right ) & =-\sum _{n=1}^{\infty }n^{2}B_{n}\left ( t\right ) X_{n}\left ( x\right ) +\sum _{n=1}^{\infty }q_{n}\left ( t\right ) X_{n}\left ( x\right ) \nonumber \\ B_{n}^{\prime }\left ( t\right ) X_{n}\left ( x\right ) +n^{2}B_{n}\left ( t\right ) X_{n}\left ( x\right ) & =q_{n}\left ( t\right ) X_{n}\left ( x\right ) \nonumber \\ B_{n}^{\prime }\left ( t\right ) +n^{2}B_{n}\left ( t\right ) & =q_{n}\left ( t\right ) \tag {1}\end{align}
This is first order ODE which is now solved for \(B_{n}\) . But first \(q_{n}\left ( t\right ) \) is found. Since
\[ t\left ( \pi -x\right ) =\sum _{n=1}^{\infty }q_{n}\left ( t\right ) X_{n}\left ( x\right ) \]
Applying orthogonality. Integrating both sides and multiplying by \(X_{m}\left ( x\right ) \) gives
\begin{align*} \int _{0}^{\pi }t\left ( \pi -x\right ) X_{m}\left ( x\right ) dx & =\sum _{n=1}^{\infty }q_{n}\left ( t\right ) \int _{0}^{\pi }X_{n}\left ( x\right ) X_{m}\left ( x\right ) dx\\ & =\sum _{n=1}^{\infty }q_{n}\left ( t\right ) \int _{0}^{\pi }\sin \left ( nx\right ) \sin \left ( mx\right ) dx\\ & =q_{m}\left ( t\right ) \int _{0}^{\pi }\sin ^{2}\left ( mx\right ) dx\\ & =\frac {\pi }{2}q_{m}\left ( t\right ) \end{align*}
Therefore
\begin{align*} q_{n}\left ( t\right ) & =\frac {2}{\pi }\int _{0}^{\pi }t\left ( \pi -x\right ) \sin \left ( nx\right ) dx\\ & =\frac {2t\left ( \pi n-\sin \left ( n\pi \right ) \right ) }{\pi n^{2}}\end{align*}
But \(n\) is integer, hence the above simplifies to
\[ q_{n}\left ( t\right ) =\frac {2t}{n}\]
Hence (1) becomes
\[ B_{n}^{\prime }\left ( t\right ) +n^{2}B_{n}\left ( t\right ) =\frac {2t}{n}\]
Integrating factor is \(I=e^{\int n^{2}dt}=e^{n^{2}t}\) . The above becomes
\[ \frac {d}{dt}\left ( B_{n}e^{n^{2}t}\right ) =\frac {2t}{n}e^{n^{2}t}\]
Integrating
\begin{align} B_{n}e^{n^{2}t} & =\frac {2}{n}\int _{0}^{t}se^{n^{2}s}ds+C\nonumber \\ B_{n}\left ( t\right ) & =\frac {2}{n}e^{-n^{2}t}\int _{0}^{t}se^{n^{2}s}ds+Ce^{-n^{2}t}\tag {2}\end{align}
At \(t=0,B_{n}\left ( 0\right ) =0\) . The above reduces to
\[ 0=C \]
Therefore (2) becomes
\begin{align*} B_{n}\left ( t\right ) & =\frac {2}{n}e^{-n^{2}t}\int _{0}^{t}se^{n^{2}s}ds\\ & =\frac {2}{n}\int _{0}^{t}se^{n^{2}\left ( s-t\right ) }ds\\ & =\frac {2\left ( n^{2}t+e^{-n^{2}t}-1\right ) }{n^{5}}\end{align*}
Hence Eq. (A) becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\frac {2\left ( n^{2}t+e^{-n^{2}t}-1\right ) }{n^{5}}\sin \left ( nx\right ) \]
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3.1.1.16 problem number 165
Taken from Maple PDE help pages
Solve the heat equation for \(u(x,t)\)
\[ u_t= k u_{xx} + Q(x,t) \]
For \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(1,t) &= 0 \\ \end{align*}
Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Q[x, t];
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t, Assumptions -> {k > 0, t > 0, x > 0, x < 1}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\sqrt {2} \left (\int _0^t e^{-k \pi ^2 K[1]^2 (t-K[2])} \text {Integrate}\left [\sqrt {2} Q(x,K[2]) \sin (\pi x K[1]),\{x,0,1\},\text {Assumptions}\to k>0\land t>0\land x>0\land x<1\right ] \, dK[2]+e^{-k \pi ^2 t K[1]^2} \int _0^1 \sqrt {2} f(x) \sin (\pi x K[1]) \, dx\right ) \sin (\pi x K[1])\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t), t) = k*( diff ( u ( x , t), x, x))+Q(x, t);
bc := u(0, t) = 0, u(l, t) = 0;
ic := u(x, 0) = f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, bc,ic], u(x, t)) assuming 0 <= x and x <= l and k>0 and t>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi x}{l}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{l^{2}}} \int _{0}^{l}f \left (\tau \right ) \sin \left (\frac {n \pi \tau }{l}\right )d \tau }{l}\right )+2 \int _{0}^{t}\left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\sin \left (\frac {n \pi x}{l}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} \left (t -\tau \right )}{l^{2}}} \int _{0}^{l}Q \left (x , \tau \right ) \sin \left (\frac {n \pi x}{l}\right )d x}{l}\right )d \tau \]
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3.1.1.17 problem number 166
Added April 28, 2019.
Taken from https://mathematica.stackexchange.com/questions/197155/solving-a-heat-equation-problem
Solve \(u(x,t)\)
\[ u_t = u_{xx} -9 u_x \]
For \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0\\ u(1,t) &= 0 \end{align*}
Initial condition \(u(x,0)=e^\frac {45}{10}( 5 \sin (\pi x)+ 9 \sin (2 \pi x)+2 \sin (3 \pi x)) \)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == D [u[x, t], {x, 2}] - 9* D [u[x, t], x];
ic = u[x, 0] == Exp [45/10 x]*(5 Sin [ Pi *x] + 9 Sin [2* Pi *x] + 2 Sin [3* Pi *x]);
bc = {u[0, t] == 0, u[1, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to e^{-\frac {9}{4} \left (\left (9+4 \pi ^2\right ) t-2 x\right )} \sin (\pi x) \left (18 e^{5 \pi ^2 t} \cos (\pi x)+5 e^{8 \pi ^2 t}+4 \cos (2 \pi x)+2\right )\right \}\right \}\]
Maple ✗
restart ; pde := diff ( u ( x , t ), t )= diff ( u ( x , t), x$2) - 9* diff ( u ( x , t ), x ); bc := u(0,t)=0,u(1,t)=0;
ic := u(x, 0) = exp (45/10* x )*(5* sin ( Pi * x ) + 9* sin (2* Pi * x ) + 2* sin (3* Pi * x )); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve({pde, ic, bc}, u(x, t))),output= ' realtime ' ));
time expired
Hand
solution
Solve
\[ u_{t}=u_{xx}-9u_{x}\]
IC
\[ u\left ( x,0\right ) =e^{\frac {45}{10}x}\left ( 5\sin \left ( \pi x\right ) +9\sin \left ( 2\pi x\right ) +2\sin \left ( 3\pi x\right ) \right ) \]
BC
\begin{align*} u\left ( 0,x\right ) & =0\\ u\left ( 1,x\right ) & =0 \end{align*}
Let \(u=X\left ( x\right ) T\left ( t\right ) \) , the PDE becomes
\begin{align*} T^{\prime }X & =X^{\prime \prime }T-9X^{\prime }T\\ \frac {T^{\prime }}{T} & =\frac {X^{\prime \prime }}{X}-9\frac {X^{\prime }}{X}=-\lambda \end{align*}
Where \(\lambda \) is the seperation constant. From B.C. we know \(\lambda >0\) . Hence the eigenvalue ODE is
\[ X^{\prime \prime }-9X^{\prime }+\lambda X=0 \]
The solution to the above is
\begin{align*} X\left ( x\right ) & =C_{1}e^{\frac {1}{2}\left ( 9-\sqrt {81-4\lambda }\right ) x}+C_{2}e^{\frac {1}{2}\left ( 9+\sqrt {81-4\lambda }\right ) x}\\ & =e^{\frac {9x}{2}}\left ( C_{1}e^{-\frac {1}{2}\sqrt {81-4\lambda }x}+C_{2}e^{\frac {1}{2}\sqrt {81-4\lambda }x}\right ) \end{align*}
At \(X\left ( 0\right ) =0\) this gives
\[ 0=C_{1}+C_{2}\]
And at \(X\left ( 1\right ) =0\)
\begin{align*} 0 & =e^{\frac {9}{2}}\left ( C_{1}e^{-\frac {1}{2}\sqrt {81-4\lambda }}+C_{2}e^{\frac {1}{2}\sqrt {81-4\lambda }}\right ) \\ 0 & =e^{\frac {9}{2}}\left ( C_{1}e^{-\frac {1}{2}\sqrt {81-4\lambda }}-C_{1}e^{\frac {1}{2}\sqrt {81-4\lambda }}\right ) \\ 0 & =C_{1}e^{\frac {9}{2}}\left ( e^{-\frac {1}{2}\sqrt {81-4\lambda }}-e^{\frac {1}{2}\sqrt {81-4\lambda }}\right ) \end{align*}
For nontrivial solution we want
\begin{align} e^{-\frac {1}{2}\sqrt {81-4\lambda }}-e^{\frac {1}{2}\sqrt {81-4\lambda }} & =0\nonumber \\ e^{-\frac {1}{2}\sqrt {81-4\lambda }} & =e^{\frac {1}{2}\sqrt {81-4\lambda }} \tag {1}\end{align}
Case \(81-4\lambda >0\)
This means \(81-4\lambda \) must be zero or
\[ \lambda =\frac {81}{4}\]
But using this eigenvalue makes the eigenfunction zero as shown below
\begin{align*} X\left ( x\right ) & =e^{\frac {9x}{2}}\left ( C_{1}e^{-\frac {1}{2}\sqrt {81-4\lambda }x}-C_{1}e^{\frac {1}{2}\sqrt {81-4\lambda }x}\right ) \\ & =C_{1}e^{\frac {9x}{2}}\left ( e^{-\frac {1}{2}\sqrt {81-4\lambda }x}-e^{\frac {1}{2}\sqrt {81-4\lambda }x}\right ) \\ & =C_{1}e^{\frac {9x}{2}}\left ( 1-1\right ) \\ & =0 \end{align*}
Therefore \(\lambda =\frac {81}{4}\) can not be used as eigenfunction.
Case \(81-4\lambda <0\)
Then (1) becomes
\begin{align*} e^{-\frac {i}{2}\sqrt {4\lambda -81}} & =e^{\frac {i}{2}\sqrt {4\lambda -81}}\\ \cos \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) -i\sin \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) & =\cos \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) +i\sin \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) \\ 2i\sin \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) & =0\\ \sin \left ( \frac {1}{2}\sqrt {4\lambda -81}\right ) & =0\\ \frac {1}{2}\sqrt {4\lambda -81} & =n\pi \qquad n=1,2,\cdots \end{align*}
Hence
\begin{align*} \frac {1}{4}\left ( 4\lambda -81\right ) & =n^{2}\pi ^{2}\\ 4\lambda & =81+4n^{2}\pi ^{2}\\ \lambda _{n} & =\frac {81}{4}+n^{2}\pi ^{2}\end{align*}
The corresponding eigenfunctions are (and since \(C_{2}=-C_{1}\) ) then
\begin{align*} X_{n}\left ( x\right ) & =C_{n}\left ( e^{\frac {1}{2}\left ( 9-\sqrt {81-4\lambda _{n}}\right ) x}-e^{\frac {1}{2}\left ( 9+\sqrt {81-4\lambda _{n}}\right ) x}\right ) \\ & =C_{n}\left ( e^{\frac {1}{2}\left ( 9-i\sqrt {4\lambda _{n}-81}\right ) x}-e^{\frac {1}{2}\left ( 9+i\sqrt {4\lambda _{n}-81}\right ) x}\right ) \\ & =C_{n}e^{\frac {9x}{2}}\left ( e^{-\frac {i}{2}\sqrt {4\lambda _{n}-81}x}-e^{\frac {i}{2}\sqrt {4\lambda _{n}-81}x}\right ) \\ & =C_{n}e^{\frac {9x}{2}}\left ( \cos \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) -i\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) -\cos \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) -\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) \right ) \\ & =C_{n}e^{\frac {9x}{2}}\left ( -2i\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) \right ) \\ & =A_{n}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) \end{align*}
Hence the solution is
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }X_{n}\left ( t\right ) T_{n}\left ( t\right ) \]
But \(T^{\prime }+\lambda _{n}T=0\) has solution \(T=e^{-\lambda _{n}t}\) . Therefore the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }A_{n}e^{-\lambda _{n}t}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) \]
At \(t=0\)
\[ e^{\frac {45}{10}x}\left ( 5\sin \left ( \pi x\right ) +9\sin \left ( 2\pi x\right ) +2\sin \left ( 3\pi x\right ) \right ) =\sum _{n=1}^{\infty }A_{n}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{n}-81}x\right ) \]
But \(\lambda _{n}=\frac {81}{4}+n^{2}\pi ^{2}\) . The above becomes
\begin{align*} e^{\frac {45}{10}x}\left ( 5\sin \left ( \pi x\right ) +9\sin \left ( 2\pi x\right ) +2\sin \left ( 3\pi x\right ) \right ) & =A_{1}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\left ( \frac {81}{4}+\pi ^{2}\right ) -81}x\right ) \\ & +A_{2}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\left ( \frac {81}{4}+4\pi ^{2}\right ) -81}x\right ) \\ & +A_{3}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\left ( \frac {81}{4}+9\pi ^{2}\right ) -81}x\right ) \\ & +\cdots \end{align*}
Or
\begin{align*} e^{\frac {45}{10}x}\left ( 5\sin \left ( \pi x\right ) +9\sin \left ( 2\pi x\right ) +2\sin \left ( 3\pi x\right ) \right ) & =A_{1}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\pi ^{2}}x\right ) \\ & +A_{2}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {16\pi ^{2}}x\right ) \\ & +A_{3}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {36\pi ^{2}}x\right ) \\ & +\cdots \end{align*}
Or
\begin{align*} e^{\frac {45}{10}x}\left ( 5\sin \left ( \pi x\right ) +9\sin \left ( 2\pi x\right ) +2\sin \left ( 3\pi x\right ) \right ) & =A_{1}e^{\frac {9x}{2}}\sin \left ( \pi x\right ) \\ & +A_{2}e^{\frac {9x}{2}}\sin \left ( 2\pi x\right ) \\ & +A_{3}e^{\frac {9x}{2}}\sin \left ( 3\pi x\right ) \\ & +\cdots \end{align*}
By comparing coefficients, we see that \(A_{1}e^{\frac {9x}{2}}=5e^{\frac {45}{10}x}\) or \(A_{1}=e^{\left ( \frac {45}{10}-\frac {9}{2}\right ) x}=5\) and \(A_{2}=9\) and \(A_{3}=2\) and and all other \(A_{n}\) for \(n>3\) are zero. Hence the solution becomes
\begin{align*} u\left ( x,t\right ) & =5e^{-\lambda _{1}t}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{1}-81}x\right ) +9e^{-\lambda _{2}t}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{2}-81}x\right ) +2e^{-\lambda _{3}t}e^{\frac {9x}{2}}\sin \left ( \frac {1}{2}\sqrt {4\lambda _{3}-81}x\right ) \\ & =e^{-\left ( \frac {81}{4}+\pi ^{2}\right ) t}e^{\frac {9x}{2}}\sin \left ( \pi x\right ) +e^{-\left ( \frac {81}{4}+4\pi ^{2}\right ) t}e^{\frac {9x}{2}}\sin \left ( 2\pi x\right ) +e^{-\left ( \frac {81}{4}+9\pi ^{2}\right ) t}e^{\frac {9x}{2}}\sin \left ( 3\pi x\right ) \\ & =e^{-\frac {81}{4}t+\frac {9}{2}x}\left ( 5e^{-\pi ^{2}t}\sin \left ( \pi x\right ) +9e^{-4\pi ^{2}t}\sin \left ( 2\pi x\right ) +2e^{-9\pi ^{2}t}\sin \left ( 3\pi x\right ) \right ) \end{align*}
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3.1.1.18 problem number 167
Added May 5, 2019.
Solve \(u(x,t)\)
\[ u_t = x^2 u_{xx} + x u_x \]
For \(1<x<b\) and \(t>0\) . The boundary conditions are
\begin{align*} u(1,t) &= 0\\ u(b,t) &= 0 \end{align*}
Initial condition \(u(x,0)=f(x) \)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == x^2* D [u[x, t], {x, 2}] + x* D [u[x, t], x];
ic = u[x, 0] == f[x];
bc = {u[1, t] == 0, u[b, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions -> b > 1], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 e^{-\frac {\pi ^2 t K[1]^2}{\log ^2(b)}} \left (\int _1^b \frac {f(x) \sin \left (\frac {\pi K[1] \log (x)}{\log (b)}\right )}{x} \, dx\right ) \sin \left (\frac {\pi K[1] \log (x)}{\log (b)}\right )}{\log (b)}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u(x,t),t)= x^2* diff ( u ( x , t ), x$2 )+ x * diff ( u ( x , t ), x ); bc := u(1,t)=0,u(b,t)=0;
ic := u(x,0)=f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve({pde, ic, bc}, u(x, t)) assuming b>1 ),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\sin \left (\frac {n \pi \ln \left (x \right )}{\ln \left (b \right )}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{\ln \left (b \right )^{2}}} \int _{1}^{b}\frac {f \left (x \right ) \sin \left (\frac {n \pi \ln \left (x \right )}{\ln \left (b \right )}\right )}{x}d x \right )}{\ln \left (b \right )}\]
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problem number 168
Added December 29, 2018.
Solve for \(u(x,t)\) in
\[ u_t = k u_{xx} + r u \]
with \(k=\frac {1}{10},r=1\) and \(0<x<1\) and \(t>0\) . With boundary conditions
\begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}
And initial conditions \(u(x,0)=1\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; k = 1/10;
r = 1;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + r*u[x, t];
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == 1;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}-\frac {2 \left (-1+(-1)^{K[1]}\right ) e^{t-\frac {1}{10} \pi ^2 t K[1]^2} \sin (\pi x K[1])}{\pi K[1]}\right \}\right \}\]
Maple ✓
restart ; k :=1/10; r :=1; pde := diff ( u ( x , t ), t) = k* diff ( u ( x , t), x$2) + r*u(x,t);
bc := u(0,t)=0,u(1,t)=0;
ic := u(x,0) = 1;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc,ic], u(x,t))),output= ' realtime ' ));
\[u \left (x , t\right ) = -\frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {{\mathrm e}^{-\frac {t \left (\pi ^{2} n^{2}-10\right )}{10}} \left (\left (-1\right )^{n}-1\right ) \sin \left (n \pi x \right )}{n}\right )}{\pi }\]
Hand
solution
Solution added 4/3/2019.
\begin{align*} u_{t} & =ku_{xx}+ru\qquad t>0,0<x<1\\ u\left ( 0,t\right ) & =0\\ u\left ( 1,t\right ) & =0\\ u\left ( x,0\right ) & =1 \end{align*}
Let \(u=ve^{rt}\) . Hence \(u_{t}=v_{t}e^{rt}+vre^{rt}\) and \(u_{xx}=v_{xx}e^{rt}\) . Hence the PDE becomes \(v_{t}e^{rt}+vre^{rt}=v_{xx}e^{rt}+rve^{rt}\) which simplifies to
\begin{align*} v_{t} & =kv_{xx}\qquad t>0,0<x<1\\ v\left ( 0,t\right ) & =0\\ v\left ( 1,t\right ) & =0 \end{align*}
The above is now in canonical form, it is standard heat PDE with homogeneous B.C. This has the solution
\[ v\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-n^{2}\pi ^{2}t}\sin \left ( n\pi x\right ) \]
Therefore
\begin{align*} u & =ve^{rt}\\ u & =\sum _{n=1}^{\infty }B_{n}e^{-t\left ( n^{2}\pi ^{2}-r\right ) }\sin \left ( n\pi x\right ) \end{align*}
At \(t=0\) the above becomes
\[ 1=\sum _{n=1}^{\infty }B_{n}\sin \left ( n\pi x\right ) \]
Hence \(B_{n}\) are \(\sin \) Fourier coefficient of \(1\) which is
\begin{align*} B_{n} & =2\int _{0}^{1}\sin \left ( n\pi x\right ) dx\\ & =2\left ( -\frac {1}{n\pi }\right ) \left ( \cos n\pi x\right ) _{0}^{1}\\ & =\frac {-2}{n\pi }\left ( \left ( -1\right ) ^{n}-1\right ) \end{align*}
Hence the solution becomes
\[ u=\sum _{n=1}^{\infty }\frac {-2}{n\pi }\left ( \left ( -1\right ) ^{n}-1\right ) e^{-t\left ( n^{2}\pi ^{2}-r\right ) }\sin \left ( n\pi x\right ) \]
But \(r=\frac {1}{10}\) , therefore
\[ u\left ( x,t\right ) =-\frac {2}{\pi }\sum _{n=1}^{\infty }\frac {\left ( \left ( -1\right ) ^{n}-1\right ) }{n}e^{-t\left ( n^{2}\pi ^{2}-\frac {1}{10}\right ) }\sin \left ( n\pi x\right ) \]
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problem number 169
Added December 29, 2018.
Solve for \(u(x,t)\) in
\[ u_t = k u_{xx} + r u\left (1- \frac {u}{\alpha } \right ) \]
with \(k=\frac {1}{100},r=\frac {1}{10},\alpha =10\) and \(0<x<1\) and \(t>0\) .
With boundary conditions
\begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}
And initial conditions \(u(x,0)=1\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; k = 1/100;
r = 1/10;
alpha = 10;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + r*u[x, t]*(1 - u[x, t]/alpha);
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == 1;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\{\}\]
Maple ✗
restart ; k := 1/100;
r := 1/10;
alpha := 10;
pde := diff ( u ( x , t), t)= k* diff ( u ( x , t),x$2) + r*u(x, t)*(1- u(x,t)/alpha);
bc := u(0, t) = 0, u(1, t) = 0;
ic := u(x, 0) =1;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic, bc], u(x, t))),output= ' realtime ' ));
sol=()
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problem number 170
Added December 29, 2018.
Solve for \(u(x,t)\) in
\[ u_t = k u_{xx} + \alpha u + \beta u ^2 - \gamma u^3 \]
with \(k=\frac {1}{1000},\alpha =\frac {1}{100},\beta =\frac {1}{100}, \gamma =\frac {5}{1000}\) and \(0<x<1\) and \(t>0\) .
With boundary conditions
\begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}
And initial conditions \(u(x,0)=1\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; k = 1/1000;
alpha = 1/10;
beta = 1/100;
gamma = 5/1000;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + alpha*u[x, t] + beta*u[x, t]^2 - gamma*u[x, t]^3;
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == 1;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\{\}\]
Maple ✗
restart ; k := 1/100;
alpha :=1/100; beta :=1/1000; g :=5/1000; pde := diff ( u ( x , t), t)= k* diff ( u ( x , t),x$2) +
alpha*u(x,t)+ beta*u(x,t)^2 - g*u(x,t)^3;
bc := u(0, t) = 0, u(1, t) = 0;
ic := u(x, 0) =1;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc], u(x, t))),output= ' realtime ' ));
sol=()
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3.1.1.22 problem number 171
This is problem 2.3.8, from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ u_t = k u_{xx} - \alpha u \]
This corresponds to a one-dimentional rod either with heat loss through the lateral sides with outside temperature zero degrees (\(\alpha >0\) ) or with insulated sides with a heat sink propertional to the temperature.
Suppose the boundary conditions are \(u(0,t)=0,u(L,t)=0\) , solve with the temperature initially \(u\left ( x,0\right ) =f(x)\) if \(\alpha >0\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] - alpha*u[x, t];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0, alpha > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} e^{-t \left (\frac {k \pi ^2 K[1]^2}{L^2}+\alpha \right )} \left (\int _0^L \frac {\sqrt {2} f(x) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}} \, dx\right ) \sin \left (\frac {\pi x K[1]}{L}\right )}{\sqrt {L}}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )- alpha * u ( x , t ); bc := u(0,t)=0,u(L,t)=0;
ic := u(x,0)=f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming L>0,alpha>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {{\mathrm e}^{-\frac {t \left (\pi ^{2} k \,n^{2}+\alpha \,L^{2}\right )}{L^{2}}} \sin \left (\frac {n \pi x}{L}\right ) \int _{0}^{L}f \left (x \right ) \sin \left (\frac {n \pi x}{L}\right )d x}{L}\right )\]
Hand
solution
\begin{align*} \frac {\partial u}{\partial t} & =k\frac {\partial ^{2}u}{\partial x^{2}}-\alpha u\\ \frac {\partial u}{\partial t}+\alpha u & =k\frac {\partial ^{2}u}{\partial x^{2}}\end{align*}
Assuming \(u\left ( x,t\right ) =X\left ( x\right ) T\left ( t\right ) \) and substituting in the above gives
\[ XT^{\prime }+\alpha XT=kTX^{\prime \prime }\]
Dividing by \(kXT\neq 0\)
\[ \frac {T^{\prime }}{kT}+\frac {\alpha }{k}=\frac {X^{\prime \prime }}{X}\]
Since each side depends on different independent variable and both are equal, they must be both equal to same constant, say \(-\lambda \) . Where \(\lambda \) is assumed real.
\[ \frac {1}{k}\frac {T^{\prime }}{T}+\frac {\alpha }{k}=\frac {X^{\prime \prime }}{X}=-\lambda \]
The two ODE’s are
\begin{align*} \frac {1}{k}\frac {T^{\prime }}{T}+\frac {\alpha }{k} & =-\lambda \\ \frac {X^{\prime \prime }}{X} & =-\lambda \end{align*}
Or
\begin{align*} T^{\prime }+\left ( \alpha +\lambda k\right ) T & =0\\ X^{\prime \prime }+\lambda X & =0 \end{align*}
The solution to the space ODE is the familiar (where \(\lambda >0\) is only possible case, As found in Haberman problem 2.3.3, part d. Since it has the same B.C.)
\[ X_{n}=B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \qquad n=1,2,3,\cdots \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}\) . The time ODE is now solved.
\[ \frac {dT_{n}}{dt}+\left ( \alpha +\lambda _{n}k\right ) T_{n}=0 \]
This has the solution
\begin{align*} T_{n}\left ( t\right ) & =e^{-\left ( \alpha +\lambda _{n}k\right ) t}\\ & =e^{-\alpha t}e^{-\left ( \frac {n\pi }{L}\right ) ^{2}kt}\end{align*}
For the same eigenvalues. Notice that no need to add a constant here, since it will be absorbed in the \(B_{n}\) when combined in the following step below. Therefore the solution to the PDE is
\[ u_{n}\left ( x,t\right ) =T_{n}\left ( t\right ) X_{n}\left ( x\right ) \]
But for linear system sum of eigenfunctions is also a solution. Hence
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }u_{n}\left ( x,t\right ) \\ & =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) e^{-\alpha t}e^{-\left ( \frac {n\pi }{L}\right ) ^{2}kt}\\ & =e^{-\alpha t}\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) e^{-\left ( \frac {n\pi }{L}\right ) ^{2}kt}\end{align*}
Where \(e^{-\alpha t}\) was moved outside since it does not depend on \(n\) . From initial condition
\[ u\left ( x,0\right ) =f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \]
Applying orthogonality of \(\sin \) as before to find \(B_{n}\) results in
\[ B_{n}=\frac {2}{L}\int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx \]
Hence the solution becomes
\begin{align*} u\left ( x,t\right ) & =\frac {2}{L}e^{-\alpha t}\left ( \sum _{n=1}^{\infty }\left ( \int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx\right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-\left ( \frac {n\pi }{L}\right ) ^{2}kt}\right ) \\ & =\frac {2}{L}\sum _{n=1}^{\infty }\left ( \int _{0}^{L}\sin \left ( \frac {n\pi }{L}x\right ) f\left ( x\right ) dx\right ) \sin \left ( \frac {n\pi }{L}x\right ) e^{-t\left ( \frac {n^{2}\pi ^{2}k+\alpha L^{2}}{L^{2}}\right ) }\end{align*}
Hence it is clear that in the limit as \(t\) becomes large \(u\left ( x,t\right ) \rightarrow 0\) since \(\alpha >0\) and
\[ \lim _{t\rightarrow \infty }u\left ( x,t\right ) =0 \]
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3.1.1.23 problem number 172
Solve the heat equation
\[ u_t = u_{xx} - u(x,t) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition is \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] + u[x, t] == D [u[x, t], {x, 2}];
ic = u[x, 0] == f[x];
bc = {u[0, t] == 0, u[L, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\sqrt {2} e^{t \left (-\frac {\pi ^2 K[1]^2}{L^2}-1\right )} \sqrt {\frac {1}{L}} \left (\int _0^L \sqrt {2} \sqrt {\frac {1}{L}} f(x) \sin \left (\frac {\pi x K[1]}{L}\right ) \, dx\right ) \sin \left (\frac {\pi x K[1]}{L}\right )\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )+ u ( x , t )= diff ( u ( x , t ), x$2 ); ic := u(x,0)=f(x);
bc := u(0,t)=0,u(L,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t )) assuming L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {{\mathrm e}^{-\frac {t \left (\pi ^{2} n^{2}+L^{2}\right )}{L^{2}}} \sin \left (\frac {n \pi x}{L}\right ) \int _{0}^{L}f \left (x \right ) \sin \left (\frac {n \pi x}{L}\right )d x}{L}\right )\]
Hand
solution
\begin{align*} u_{t} & =u_{xx}-u\\ u\left ( 0,t\right ) & =0\\ u\left ( L,t\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
Let \(u\left ( x,t\right ) =v\left ( x,t\right ) e^{-t}\) , hence \(u_{t}=v_{t}e^{-t}-ve^{-t}\) and \(u_{xx}=v_{xx}e^{-t}\) . Therefore the above PDE becomes
\begin{align*} v_{t}e^{-t}-ve^{-t} & =v_{xx}e^{-t}-ve^{-t}\\ v_{t} & =v_{xx}\end{align*}
With boundary conditions
\begin{align*} v\left ( 0,t\right ) & =0\\ v\left ( L,t\right ) & =0 \end{align*}
The solution to this PDE is known, since it has homogenouse BC and it is in standard form. The solution is
\[ v\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}e^{-\left ( \frac {n\pi }{L}\right ) ^{2}t}\sin \left ( \frac {n\pi }{L}x\right ) \]
Hence
\begin{align} u\left ( x,t\right ) & =e^{-t}\sum _{n=1}^{\infty }B_{n}e^{-\left ( \frac {n^{2}\pi ^{2}}{L^{2}}\right ) t}\sin \left ( \frac {n\pi }{L}x\right ) \nonumber \\ & =\sum _{n=1}^{\infty }B_{n}e^{-\left ( \frac {n^{2}\pi ^{2}}{L^{2}}\right ) t-t}\sin \left ( \frac {n\pi }{L}x\right ) \nonumber \\ & =\sum _{n=1}^{\infty }B_{n}e^{-t\left ( \frac {n^{2}\pi ^{2}}{L^{2}}+1\right ) }\sin \left ( \frac {n\pi }{L}x\right ) \nonumber \\ & =\sum _{n=1}^{\infty }B_{n}e^{-t\left ( \frac {n^{2}\pi ^{2}+L^{2}}{L^{2}}\right ) }\sin \left ( \frac {n\pi }{L}x\right ) \tag {1}\end{align}
Applying initial conditions \(u\left ( x,0\right ) =f\left ( x\right ) \) gives
\[ f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \frac {n\pi }{L}x\right ) \]
Hence \(B_{n}\) are the Fourier sine coefficients of \(f\left ( x\right ) \)
\[ B_{n}=\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx \]
Therefore the solution in (1) becomes
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }\frac {2}{L}\left ( \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-t\left ( \frac {n^{2}\pi ^{2}+L^{2}}{L^{2}}\right ) }\sin \left ( \frac {n\pi }{L}x\right ) \\ & =\frac {2}{L}\sum _{n=1}^{\infty }\sin \left ( \frac {n\pi }{L}x\right ) e^{-t\left ( \frac {n^{2}\pi ^{2}+L^{2}}{L^{2}}\right ) }\left ( \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) \end{align*}
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3.1.1.24 problem number 173
Added Feb 10, 2018.
Solve the heat equation
\[ u_t + u(x,t) = 100 u_{xx} \]
For \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(1,t) &= 0 \end{align*}
Initial condition is \(u(x,0)=\sin (2 \pi x)-\sin (5 \pi x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == 100* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, u[1, t] == 0};
ic = u[x, 0] == Sin [2* Pi *x] - Sin [5* Pi *x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to e^{-400 \pi ^2 t} \sin (2 \pi x)-e^{-2500 \pi ^2 t} \sin (5 \pi x)\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )=100* diff ( u ( x , t ), x$2 ); ic := u(x,0)= sin (2* Pi * x )- sin (5* Pi * x ); bc := u(0,t)=0,u(1,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \sin \left (2 \pi x \right ) {\mathrm e}^{-400 \pi ^{2} t}-\sin \left (5 \pi x \right ) {\mathrm e}^{-2500 \pi ^{2} t}\]
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3.1.1.25 problem number 174
Added June 23, 2019.
Solve the heat equation
\[ u_t = k u_{xx} - u x \]
For \(0<x<\pi \) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(\pi ,t) &= 0 \end{align*}
Initial condition \(u(x,0)=\sin (x)\)
Mathematica ✗
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] - u[x,t]*x;
bc = {u[0, t] == 0, u[ Pi , t] == 0};
ic = u[x, 0] == Sin [x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> k > 0], 60*10]];
Failed
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )- u(x,t)*x;
ic := u(x,0)= sin ( x ); bc := u(0,t)=0,u( Pi , t )=0; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t )) assuming k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = -\left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\left (-\operatorname {AiryAi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right ) \operatorname {AiryBi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )+\operatorname {AiryBi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right ) \operatorname {AiryAi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )\right ) \left (\operatorname {AiryAi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right ) \int _{0}^{\pi }\sin \left (x \right ) \operatorname {AiryBi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right )d x -\operatorname {AiryBi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right ) \int _{0}^{\pi }\sin \left (x \right ) \operatorname {AiryAi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right )d x \right ) \left (\sinh \left (\lambda _{n} t \right )-\cosh \left (\lambda _{n} t \right )\right )}{\int _{0}^{\pi }\operatorname {AiryBi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right )^{2}d x \operatorname {AiryAi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )^{2}-2 \operatorname {AiryBi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right ) \int _{0}^{\pi }\operatorname {AiryAi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right ) \operatorname {AiryBi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right )d x \operatorname {AiryAi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )+\operatorname {AiryBi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )^{2} \int _{0}^{\pi }\operatorname {AiryAi}\left (\frac {-\lambda _{n}+x}{k^{{1}/{3}}}\right )^{2}d x}\right )\boldsymbol {\operatorname {where}}\left \{-\operatorname {AiryAi}\left (\frac {-\lambda _{n}+\pi }{k^{{1}/{3}}}\right ) \operatorname {AiryBi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )+\operatorname {AiryBi}\left (\frac {-\lambda _{n}+\pi }{k^{{1}/{3}}}\right ) \operatorname {AiryAi}\left (-\frac {\lambda _{n}}{k^{{1}/{3}}}\right )=0\wedge -\infty \le \lambda _{n}\le \infty \right \}\]
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3.1.1.26 problem number 175
Added June 23, 2019.
Solve the heat equation
\[ u_t = k u_{xx} + a u_x \]
Where \(a>0\) . For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + a* D [u[x, t], x];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {k > 0,a>0,L>0}], 60*10]];
\[\left \{\left \{u(x,t)\to e^{-\frac {a (a t+2 x)}{4 k}} \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} \left (\int _0^L e^{\frac {a x}{2 k}} f(x) \sin \left (\frac {\pi x K[1]}{L}\right ) \, dx\right ) \sin \left (\frac {\pi x K[1]}{L}\right )}{L}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )+ a* diff ( u ( x , t ), x ); ic := u(x,0)=f(x);
bc := u(0,t)=0,u(L,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t )) assuming k>0,a>0,L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = -2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\left (\sinh \left (\frac {\left (4 \pi ^{2} k^{2} n^{2}+L^{2} a^{2}\right ) t}{4 L^{2} k}\right )-\cosh \left (\frac {\left (4 \pi ^{2} k^{2} n^{2}+L^{2} a^{2}\right ) t}{4 L^{2} k}\right )\right ) \left (\cosh \left (\frac {x a}{2 k}\right )-\sinh \left (\frac {x a}{2 k}\right )\right ) \sin \left (\frac {\pi x n}{L}\right ) \int _{0}^{L}f \left (x \right ) \left (\cosh \left (\frac {x a}{2 k}\right )-\sinh \left (\frac {x a}{2 k}\right )\right ) \sin \left (\frac {\pi x n}{L}\right ) {\mathrm e}^{\frac {x a}{k}}d x}{L}\right )\]
Hand
solution
Solve
\begin{align*} u_{t} & =ku_{xx}+au_{x}\\ u\left ( 0,t\right ) & =0\\ u\left ( L,0\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
Trying separation of variables. Let \(u=XT\) , then the PDE becomes
\begin{align*} T^{\prime }X & =kX^{\prime \prime }T+aX^{\prime }T\\ \frac {1}{k}\frac {T^{\prime }}{T} & =\frac {X^{\prime \prime }}{X}+\frac {a}{k}\frac {X^{\prime }}{X}=-\lambda \end{align*}
Where \(\lambda \) is separation constant. Hence
\[ T^{\prime }+k\lambda T=0 \]
Which has solution \(T\left ( t\right ) =e^{-k\lambda t}\) (the constant of integration is not needed, it will be combined with constant coming from the spatial ODE), and the spatial ODE is
\begin{align*} \frac {X^{\prime \prime }}{X}+\frac {a}{k}\frac {X^{\prime }}{X} & =-\lambda \\ X^{\prime \prime }+\frac {a}{k}X^{\prime }+\lambda X & =0\\ X\left ( 0\right ) & =0\\ X\left ( L\right ) & =0 \end{align*}
The characteristic equation is \(r^{2}+\frac {a}{k}r+\lambda =0\) .
Case \(\lambda =0\) Then \(r^{2}+\frac {a}{k}r=0\) or \(r\left ( r+\frac {a}{k}\right ) =0\) . Hence \(r=0\) or \(r=-\frac {a}{k}\) . So the solution is
\[ X\left ( x\right ) =c_{1}+c_{2}e^{-\frac {a}{k}x}\]
At \(X\left ( 0\right ) =0\) the above gives \(0=c_{1}+c_{2}\) . Therefore the solution is \(X\left ( x\right ) =c_{1}\left ( 1-e^{\frac {-a}{k}x}\right ) \) . At \(X\left ( L\right ) =0\) then \(c_{1}\left ( 1-e^{\frac {a}{k}L}\right ) =0\) . This means this for non-trivial solution \(e^{\frac {-a}{k}L}=1\) or \(\frac {-a}{k}L=0\) . Hence \(\lambda =0\) is not an eigenvalue
Case \(\lambda <0\) Then \(r^{2}+\frac {a}{k}r+\lambda =0\) or \(r=\frac {-b}{2}\pm \frac {\sqrt {b^{2}-4ac}}{2a}=\frac {-a}{2k}\pm \frac {\sqrt {\frac {a^{2}}{k^{2}}-4\lambda }}{2}\) . Since \(\lambda <0\) then the term under the sqrt root is positive. Hence the solution will have real roots. Not complex conjugate. Hence the solution is pure exponentials. Let \(\frac {a^{2}}{k^{2}}-4\lambda =\alpha \) and \(\frac {a}{2k}=\beta \) then
\begin{align*} r_{1} & =-\beta +\sqrt {\alpha }\\ r_{2} & =-\beta -\sqrt {\alpha }\end{align*}
Then \(X\left ( x\right ) =c_{1}\cosh \left ( \left ( -\beta +\alpha \right ) x\right ) +c_{2}\sinh \left ( \left ( -\beta -\alpha \right ) x\right ) \) . At \(X\left ( 0\right ) =0\) this gives \(0=c_{1}\) . Hence solution is \(X\left ( x\right ) =c_{2}\sinh \left ( \left ( -\beta -\alpha \right ) x\right ) \) . At \(X\left ( L\right ) =0\) , this gives
\[ 0=c_{2}\sinh \left ( \left ( -\beta -\alpha \right ) L\right ) \]
For non-trivial solution, \(\sinh \left ( \left ( -\beta -\alpha \right ) L\right ) =0\) . But this is zero only when \(\left ( -\beta -\alpha \right ) L\) or \(\beta =-\alpha \) . Or \(\frac {-a}{2k}=-\sqrt {\frac {a^{2}}{k^{2}}-4\lambda }\) . Or \(\frac {a^{2}}{4k^{2}}=\frac {a^{2}}{k^{2}}-4\lambda \) or
\begin{align*} 4\lambda & =\frac {a^{2}}{k^{2}}-\frac {a^{2}}{4k^{2}}\\ \lambda & =\frac {3}{16}\frac {a^{2}}{k^{2}}\end{align*}
But \(\lambda \) was assumed negative. Hence this is not possible. \(\lambda <0\) is not eigenvalue
Case \(\lambda >0\) Then \(r^{2}+\frac {a}{k}r+\lambda =0\) or \(r=\frac {-a}{2k}\pm \frac {1}{2}\sqrt {\frac {a^{2}}{k^{2}}-4\lambda }=\frac {-a}{2k}\pm \sqrt {\frac {a^{2}}{4k^{2}}-\lambda }\) . Since \(\lambda >0\) then the term under the sqrt root can be negative. Then only when \(\lambda >\frac {a^{2}}{4k^{2}}\) will there be complex roots. Therefore assuming \(\lambda >\frac {a^{2}}{4k^{2}}\) , then
\begin{align*} r_{1} & =\frac {-a}{2k}+i\sqrt {\lambda -\frac {a^{2}}{4k^{2}}}\\ r_{2} & =\frac {-a}{2k}-i\sqrt {\lambda -\frac {a^{2}}{4k^{2}}}\end{align*}
The solution is
\[ X\left ( x\right ) =e^{\frac {-a}{2k}x}\left ( c_{1}\cos \left ( \sqrt {\lambda -\frac {a^{2}}{4k^{2}}}x\right ) +c_{2}\sin \left ( \sqrt {\lambda -\frac {a^{2}}{4k^{2}}}x\right ) \right ) \]
At \(X\left ( 0\right ) =0\) this gives \(0=c_{1}\) . Hence solution is
\[ X\left ( x\right ) =e^{\frac {-a}{2k}x}c_{2}\sin \left ( \sqrt {\lambda -\frac {a^{2}}{4k^{2}}}x\right ) \]
At \(X\left ( L\right ) =0\) , this gives
\[ 0=e^{\frac {-a}{2k}\pi }c_{2}\sin \left ( \sqrt {\lambda -\frac {a^{2}}{4k^{2}}}L\right ) \]
For non-trivial solution, \(\sin \left ( \sqrt {\lambda -\frac {a^{2}}{4k^{2}}}L\right ) =0\) or \(\sqrt {\lambda -\frac {a^{2}}{4k^{2}}}L=n\pi \) . Hence
\begin{align*} \lambda -\frac {a^{2}}{4k^{2}} & =\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \\ \lambda _{n} & =\left ( \frac {n\pi }{L}\right ) ^{2}+\frac {a^{2}}{4k^{2}}\qquad n=1,2,3,\cdots \end{align*}
And the corresponding eigenfunction is
\[ X_{n}\left ( x\right ) =e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \qquad n=1,2,3,\cdots \]
Therefore the solution is
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }c_{n}e^{-k\lambda _{n}t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
\(c_{n}\) is found from initial conditions. At \(t=0\)
\begin{align*} f\left ( x\right ) & =\sum _{n=1}^{\infty }c_{n}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \\ e^{\frac {a}{2k}x}f\left ( x\right ) & =\sum _{n=1}^{\infty }c_{n}\sin \left ( \frac {n\pi }{L}x\right ) \end{align*}
Therefore, by orthogonality
\begin{align*} \int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx & =c_{n}\int _{0}^{L}\sin ^{2}\left ( \frac {n\pi }{L}x\right ) dx\\ \int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx & =c_{n}\frac {L}{2}\\ c_{n} & =\frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx \end{align*}
Hence the solution becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-k\lambda _{n}t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}+\frac {a^{2}}{4k^{2}}\qquad n=1,2,3,\cdots \)
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-k\left ( \frac {n^{2}\pi ^{2}}{L^{2}}+\frac {a^{2}}{4k^{2}}\right ) t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
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3.1.1.27 problem number 176
Added June 23, 2019.
Solve the heat equation
\[ u_t = k u_{xx} + a u_x \]
Where \(a>0\) . For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u(0,t) &= 0 \\ u(L,t) &= 0 \end{align*}
Initial condition \(u(x,0)=f(x)\) using the following values
\begin{align*} f(x)&=\sin (x)\\ k&=1\\ a&=5\\ L&=\pi \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; f = Sin [ x ]; L = Pi ; k =1; a =5; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + a* D [u[x, t], x];
bc = {u[0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to -\frac {-\text {Integrate}\left [\frac {\left (1+e^{-i \pi K}\right ) e^{K (-K t+5 i t+i x)}}{1-K^2},\{K,-\infty ,\infty \},\text {Assumptions}\to K\in \mathbb {C}\right ]+\int _{\infty }^{-\infty }\frac {\exp \left (i t \sqrt {4 K[1]^2+25} K[1]+\frac {1}{2} x \left (2 i K[1]+\sqrt {4 K[1]^2+25}-5\right )\right ) \left (\frac {1+e^{-\frac {1}{2} \pi \left (2 i K[1]+\sqrt {4 K[1]^2+25}-5\right )}}{\frac {1}{4} \left (2 i K[1]+\sqrt {4 K[1]^2+25}-5\right )^2+1}-\frac {1+e^{\frac {1}{2} \pi \left (2 i K[1]+\sqrt {4 K[1]^2+25}+5\right )}}{\frac {1}{4} \left (2 i K[1]+\sqrt {4 K[1]^2+25}+5\right )^2+1}\right )}{-1+e^{\pi \left (2 i K[1]+\sqrt {4 K[1]^2+25}\right )}}dK[1]+\int _{-\infty }^{\infty }\exp \left (-i t \sqrt {4 K[1]^2+25} K[1]-\frac {1}{2} x \left (-2 i K[1]+\sqrt {4 K[1]^2+25}+5\right )\right ) \left (\frac {2 \left (1+\exp \left (-\frac {1}{2} \pi \left (-2 i K[1]+\sqrt {4 K[1]^2+25}-5\right )\right )\right )}{\left (-1+e^{2 i \pi K[1]-\pi \sqrt {4 K[1]^2+25}}\right ) \left (2 i K[1] \left (\sqrt {4 K[1]^2+25}-5\right )+5 \sqrt {4 K[1]^2+25}-27\right )}+\frac {2 i e^{2 i \pi K[1]} \left (1+e^{\frac {1}{2} \pi \left (-2 i K[1]+\sqrt {4 K[1]^2+25}+5\right )}\right )}{\left (e^{2 i \pi K[1]}-e^{\pi \sqrt {4 K[1]^2+25}}\right ) \left (2 K[1] \left (\sqrt {4 K[1]^2+25}+5\right )+i \left (5 \sqrt {4 K[1]^2+25}+27\right )\right )}\right )dK[1]}{2 \pi }\right \}\right \}\]
Maple ✓
restart ; f := sin ( x ); L := Pi ; k :=1; a :=5; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )+ a* diff ( u ( x , t ), x ); ic := u(x,0)=f;
bc := u(0,t)=0,u(L,t)=0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t )) ),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {160 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {n \left (\left (-1\right )^{n} {\mathrm e}^{\frac {5 \pi }{2}}+1\right ) \sin \left (n x \right ) \left (\sinh \left (t \,n^{2}+\frac {25}{4} t \right )-\cosh \left (t \,n^{2}+\frac {25}{4} t \right )\right ) \left (\cosh \left (\frac {5 x}{2}\right )-\sinh \left (\frac {5 x}{2}\right )\right )}{16 n^{4}+168 n^{2}+841}\right )}{\pi }\]
Hand
solution
Solve
\begin{align*} u_{t} & =ku_{xx}+au_{x}\\ u\left ( 0,t\right ) & =0\\ u\left ( L,0\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
With \(a=5,k=1,L=\pi ,f\left ( x\right ) =\sin \left ( x\right ) \) . The above problem was solved in 3.1.1.26 on page 431 and the solution is
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-k\lambda _{n}t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
Where \(\lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}+\frac {a^{2}}{4k^{2}}\qquad n=1,2,3,\cdots \)
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}f\left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-k\left ( \frac {n^{2}\pi ^{2}}{L^{2}}+\frac {a^{2}}{4k^{2}}\right ) t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
Substituting for \(f\left ( x\right ) =\sin \left ( x\right ) \) the above becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {2}{L}\int _{0}^{L}e^{\frac {a}{2k}x}\sin \left ( x\right ) \sin \left ( \frac {n\pi }{L}x\right ) dx\right ) e^{-k\left ( \frac {n^{2}\pi ^{2}}{L^{2}}+\frac {a^{2}}{4k^{2}}\right ) t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
But
\[ \int _{0}^{\pi }e^{\frac {a}{2k}x}\sin \left ( x\right ) \sin \left ( nx\right ) dx=\frac {-16nak^{3}\left ( 1+\left ( -1\right ) ^{n}e^{\frac {\pi a}{2k}}\right ) }{a^{4}+16k^{4}\left ( n^{2}-1\right ) ^{2}+8a^{2}k^{2}\left ( 1+n^{2}\right ) }\]
Hence the solution becomes
\[ u\left ( x,t\right ) =\frac {-32ak^{3}}{L}\sum _{n=1}^{\infty }\left ( \frac {n\left ( 1+\left ( -1\right ) ^{n}e^{\frac {\pi a}{2k}}\right ) }{a^{4}+16k^{4}\left ( n^{2}-1\right ) ^{2}+8a^{2}k^{2}\left ( 1+n^{2}\right ) }\right ) e^{-k\left ( \frac {n^{2}\pi ^{2}}{L^{2}}+\frac {a^{2}}{4k^{2}}\right ) t}e^{\frac {-a}{2k}x}\sin \left ( \frac {n\pi }{L}x\right ) \]
But \(L=\pi \) , hence
\[ u\left ( x,t\right ) =\frac {-32ak^{3}}{\pi }\sum _{n=1}^{\infty }\left ( \frac {n\left ( 1+\left ( -1\right ) ^{n}e^{\frac {\pi a}{2k}}\right ) }{a^{4}+16k^{4}\left ( n^{2}-1\right ) ^{2}+8a^{2}k^{2}\left ( 1+n^{2}\right ) }\right ) e^{-k\left ( n^{2}+\frac {a^{2}}{4k^{2}}\right ) t}e^{\frac {-a}{2k}x}\sin \left ( nx\right ) \]
And \(a=5\)
\[ u\left ( x,t\right ) =\frac {-160k^{3}}{\pi }\sum _{n=1}^{\infty }\left ( \frac {n\left ( 1+\left ( -1\right ) ^{n}e^{\frac {5\pi }{2k}}\right ) }{625+16k^{4}\left ( n^{2}-1\right ) ^{2}+200k^{2}\left ( 1+n^{2}\right ) }\right ) e^{-k\left ( n^{2}+\frac {25}{5k^{2}}\right ) t}e^{\frac {-5}{2k}x}\sin \left ( nx\right ) \]
And \(k=1\)
\[ u\left ( x,t\right ) =\frac {-160}{\pi }\sum _{n=1}^{\infty }\left ( \frac {n\left ( 1+\left ( -1\right ) ^{n}e^{\frac {5\pi }{2}}\right ) }{625+16\left ( n^{2}-1\right ) ^{2}+200\left ( 1+n^{2}\right ) }\right ) e^{-\left ( n^{2}+\frac {25}{4}\right ) t}e^{\frac {-5}{2}x}\sin \left ( nx\right ) \]
Animation is below
Source code used for the above
The following animations for \(a=1,a=7\) . Showing the effect of increasing the convection factor \(a\) on the solution.
\(a=1\)
\(a=7\)
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3.1.1.28 problem number 177
This is problem 2.4.2 from Richard Haberman applied partial differential equations, 5th edition.
Solve the heat equation
\[ u_t = k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u(L,t)=0\) with the temperature initially \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0 && k > 0}], 60*10]];
sol = sol /. { K [1] -> n, K [2] -> x};
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=0}{\overset {\infty }{\sum }}e^{-\frac {k (2 \pi n+\pi )^2 t}{4 L^2}} \cos \left (\frac {(2 n+1) \pi x}{2 L}\right ) \int _0^L \cos \left (\frac {(2 n+1) \pi x}{2 L}\right ) f(x) \, dx}{L}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, u ( L , t )=0; ic := u(x,0)=f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming L>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\cos \left (\frac {\left (1+2 n \right ) \pi x}{2 L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} \left (1+2 n \right )^{2} t}{4 L^{2}}} \int _{0}^{L}f \left (x \right ) \cos \left (\frac {\left (1+2 n \right ) \pi x}{2 L}\right )d x}{L}\right )\]
Hand
solution
Solve
\[ \frac {\partial u}{\partial t}=\kappa \frac {\partial ^{2}u}{\partial x^{2}}\]
Let \(u\left ( x,t\right ) =T\left ( t\right ) X\left ( x\right ) \) , then the PDE becomes
\[ \frac {1}{\kappa }T^{\prime }X=X^{\prime \prime }T \]
Dividing by \(XT\)
\[ \frac {1}{\kappa }\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}\]
Since each side depends on different independent variable and both are equal, they must be both equal to same constant, say \(-\lambda \) . Where \(\lambda \) is real.
\[ \frac {1}{\kappa }\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}=-\lambda \]
The two ODE’s are
\begin{align} T^{\prime }+k\lambda T & =0\tag {1}\\ X^{\prime \prime }+\lambda X & =0\tag {2}\end{align}
Per problem statement, \(\lambda \geq 0\) , so only two cases needs to be examined.
Case \(\lambda =0\)
The space equation becomes \(X^{\prime \prime }=0\) with the solution
\[ X=Ax+b \]
Hence left B.C. implies \(X^{\prime }\left ( 0\right ) =0\) or \(A=0\) . Therefore the solution becomes \(X=b\) . The right B.C. implies \(X\left ( L\right ) =0\) or \(b=0\) . Therefore this leads to \(X=0\) as the only solution. This results in trivial solution. Therefore \(\lambda =0\) is not an eigenvalue
Case \(\lambda >0\)
Starting with the space ODE, the solution is
\begin{align*} X\left ( x\right ) & =A\cos \left ( \sqrt {\lambda }x\right ) +B\sin \left ( \sqrt {\lambda }x\right ) \\ X^{\prime }\left ( x\right ) & =-A\sqrt {\lambda }\sin \left ( \sqrt {\lambda }x\right ) +B\sqrt {\lambda }\cos \left ( \sqrt {\lambda }x\right ) \end{align*}
Left B.C. gives
\begin{align*} 0 & =X^{\prime }\left ( 0\right ) \\ & =B\sqrt {\lambda }\end{align*}
Hence \(B=0\) since it is assumed \(\lambda \neq 0\) and \(\lambda >0\) . Solution becomes
\[ X\left ( x\right ) =A\cos \left ( \sqrt {\lambda }x\right ) \]
Applying right B.C. gives
\begin{align*} 0 & =X\left ( L\right ) \\ & =A\cos \left ( \sqrt {\lambda }L\right ) \end{align*}
\(A=0\) leads to trivial solution. Therefore \(\cos \left ( \sqrt {\lambda }L\right ) =0\) or
\begin{align*} \sqrt {\lambda } & =\frac {n\pi }{2L}\qquad n=1,3,5,\cdots \\ & =\frac {\left ( 2n-1\right ) \pi }{2L}\qquad n=1,2,3\cdots \end{align*}
Hence
\begin{align*} \lambda _{n} & =\left ( \frac {n\pi }{2L}\right ) ^{2}\qquad n=1,3,5,\cdots \\ & =\frac {\left ( 2n-1\right ) ^{2}\pi ^{2}}{4L^{2}}\qquad n=1,2,3\cdots \end{align*}
Therefore
\[ X_{n}\left ( x\right ) =A_{n}\cos \left ( \frac {n\pi }{2L}x\right ) \qquad n=1,3,5,\cdots \]
And the corresponding time solution
\[ T_{n}=e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\qquad n=1,3,5,\cdots \]
Hence
\begin{align*} u_{n}\left ( x,t\right ) & =X_{n}T_{n}\\ u\left ( x,t\right ) & =\sum _{n=1,3,5,\cdots }^{\infty }A_{n}\cos \left ( \frac {n\pi }{2L}x\right ) e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\\ & =\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {\left ( 2n-1\right ) \pi }{2L}x\right ) e^{-k\left ( \frac {\left ( 2n-1\right ) \pi }{2L}\right ) ^{2}t}\end{align*}
From initial conditions
\[ f\left ( x\right ) =\sum _{n=1,3,5,\cdots }^{\infty }A_{n}\cos \left ( \frac {n\pi }{2L}x\right ) \]
Multiplying both sides by \(\cos \left ( \frac {m\pi }{2L}x\right ) \) and integrating
\[ \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{2L}x\right ) dx=\int \left ( \sum _{n=1,3,5,\cdots }^{\infty }A_{n}\cos \left ( \frac {m\pi }{2L}x\right ) \cos \left ( \frac {n\pi }{2L}x\right ) \right ) dx \]
Interchanging order of summation and integration and applying orthogonality results in
\begin{align*} \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{2L}x\right ) dx & =A_{m}\frac {L}{2}\\ A_{n} & =\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {n\pi }{2L}x\right ) dx \end{align*}
Therefore the solution is
\begin{align*} u\left ( x,t\right ) & =\frac {2}{L}\sum _{n=1,3,5,\cdots }^{\infty }\left [ \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {n\pi }{2L}x\right ) dx\right ] \cos \left ( \frac {n\pi }{2L}x\right ) e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\\ & =\frac {2}{L}\sum _{n=0}^{\infty }\left ( \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {\left ( 2n+1\right ) \pi }{2L}x\right ) dx\right ) \cos \left ( \frac {\left ( 2n+1\right ) \pi }{2L}x\right ) e^{-k\left ( \frac {\left ( 2n+1\right ) \pi }{2L}\right ) ^{2}t}\end{align*}
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3.1.1.29 problem number 178
Added January 13, 2020.
Problem 4.1.4, Introduction to Partial Differential Equations by Peter Olver ISBN 9783319020983.
Solve the heat equation
\[ u_t = k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u_x(L,t)=0\) with the temperature initially \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = {u[0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0 && k > 0}], 60*10]];
sol = sol /. { K [1] -> n, K [2] -> x};
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}e^{-\frac {k (1-2 n)^2 \pi ^2 t}{4 L^2}} \left (\int _0^L f(x) \sin \left (\frac {(2 n-1) \pi x}{2 L}\right ) \, dx\right ) \sin \left (\frac {(2 n-1) \pi x}{2 L}\right )}{L}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := u (0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)=f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming L>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\sin \left (\frac {\left (1+2 n \right ) \pi x}{2 L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} \left (1+2 n \right )^{2} t}{4 L^{2}}} \int _{0}^{L}f \left (x \right ) \sin \left (\frac {\left (1+2 n \right ) \pi x}{2 L}\right )d x}{L}\right )\]
Hand
solution
The problem to solve is to solve for \(u\left ( x,t\right ) \) in
\[ u_{t}=ku_{xx}\qquad 0<x<L,t>0 \]
With boundary conditions
\begin{align*} u\left ( 0,t\right ) & =0\\ u_{x}\left ( L,t\right ) & =0 \end{align*}
And initial conditions
\[ u\left ( x,0\right ) =f\left ( x\right ) \]
Let \(u\left ( x,t\right ) =T\left ( t\right ) X\left ( x\right ) \) , then the PDE becomes
\[ T^{\prime }X=kX^{\prime \prime }T \]
Dividing by \(XT\)
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}\]
Since each side depends on different independent variable and both are equal, they must be both equal to same constant, say \(-\lambda \) . Where \(\lambda \) is real.
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}=-\lambda \]
The two ODE’s are
\begin{equation} T^{\prime }+\lambda kT=0\tag {1}\end{equation}
\begin{align} X^{\prime \prime }+\lambda X & =0\tag {2}\\ X\left ( 0\right ) & =0\nonumber \\ X^{\prime }\left ( L\right ) & =0\nonumber \end{align}
Now we solve (2) to find the eigenvalues and eigenfunctions.
Case \(\lambda <0\)
Let \(-\lambda =\omega ^{2}\) . Hence the ODE is \(X^{\prime \prime }-\omega ^{2}X=0\) and the solution becomes
\[ X\left ( x\right ) =C_{1}\cosh \left ( \omega x\right ) +C_{2}\sinh \left ( \omega x\right ) \]
At \(x=0\) the above gives
\[ 0=C_{1}\]
Hence the solution now becomes
\[ X\left ( x\right ) =C_{2}\sinh \left ( \omega x\right ) \]
Taking derivative gives
\[ X^{\prime }\left ( x\right ) =\omega C_{2}\sinh \left ( \omega x\right ) \]
At \(x=L\)
\[ 0=\omega C_{2}\cosh \left ( \omega L\right ) \]
But \(\cosh \left ( \omega L\right ) \) is never zero. Therefore \(C_{2}=0\) which leads to trivial solution. Therefore \(\lambda <0\) is not eigenvalue .
Case \(\lambda =0\)
The space equation becomes \(X^{\prime \prime }=0\) with the solution
\[ X=Ax+B \]
At \(x=0\) the above gives \(0=B\) . Therefore the solution is \(X=Ax\) . Taking derivative gives \(X^{\prime }=A\) . At \(x=L\) this gives \(0=A\) . Which leads to trivial solutions. Therefore \(\lambda =0\) is not an eigenvalue
Case \(\lambda >0\)
Starting with the space ODE, the solution is
\[ X\left ( x\right ) =A\cos \left ( \sqrt {\lambda }x\right ) +B\sin \left ( \sqrt {\lambda }x\right ) \]
Left B.C. gives
\[ 0=A \]
The solution becomes
\[ X\left ( x\right ) =B\sin \left ( \sqrt {\lambda }x\right ) \]
Taking derivative gives
\[ X^{\prime }\left ( x\right ) =\sqrt {\lambda }B\cos \left ( \sqrt {\lambda }x\right ) \]
Applying right B.C. gives
\[ 0=\sqrt {\lambda }B\cos \left ( \sqrt {\lambda }L\right ) \]
For non trivial solution we want \(\cos \left ( \sqrt {\lambda }L\right ) =0\) or
\[ \sqrt {\lambda }=\frac {n\pi }{2L}\qquad n=1,3,5,\cdots \]
Hence the eigenvalues are
\[ \lambda _{n}=\left ( \frac {n\pi }{2L}\right ) ^{2}\qquad n=1,3,5,\cdots \]
Therefore the eigenfunctions are
\[ X_{n}\left ( x\right ) =\sin \left ( \frac {n\pi }{2L}x\right ) \qquad n=1,3,5,\cdots \]
Now that we found the eigenvalues, we can solve the time ODE (1).
\begin{align*} T_{n}^{\prime }+k\lambda _{n}T & =0\\ T_{n} & =B_{n}e^{-k\lambda _{n}t}\\ & =B_{n}e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\end{align*}
Hence the fundamental solution is
\begin{align} u_{n}\left ( x,t\right ) & =X_{n}T_{n}\nonumber \\ u\left ( x,t\right ) & =\sum _{n=1,3,5,\cdots }^{\infty }B_{n}\sin \left ( \frac {n\pi }{2L}x\right ) e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\tag {3}\end{align}
From initial conditions
\[ f\left ( x\right ) =\sum _{n=1,3,5,\cdots }^{\infty }B_{n}\sin \left ( \frac {n\pi }{2L}x\right ) \]
Multiplying both sides by \(\sin \left ( \frac {m\pi }{2L}x\right ) \) and integrating
\[ \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {m\pi }{2L}x\right ) dx=\int _{0}^{L}\left ( \sum _{n=1,3,5,\cdots }^{\infty }B_{n}\sin \left ( \frac {m\pi }{2L}xx\right ) \sin \left ( \frac {n\pi }{2L}x\right ) \right ) dx \]
Interchanging order of summation and integration and applying orthogonality between \(\cos \) functions results in
\begin{align*} \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {m\pi }{2L}x\right ) dx & =\int _{0}^{L}B_{m}\sin ^{2}\left ( \frac {m\pi }{2L}x\right ) dx\\ & =B_{m}\frac {L}{2}\end{align*}
Therefore
\[ B_{n}=\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{2L}x\right ) dx \]
Therefore the solution is (3) becomes
\begin{align*} u\left ( x,t\right ) & =\frac {2}{L}\sum _{n=1,3,5,\cdots }^{\infty }\left ( \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {n\pi }{2L}x\right ) dx\right ) \sin \left ( \frac {n\pi }{2L}x\right ) e^{-k\left ( \frac {n\pi }{2L}\right ) ^{2}t}\\ & =\frac {2}{L}\sum _{n=0}^{\infty }\left ( \int _{0}^{L}f\left ( x\right ) \sin \left ( \frac {\left ( 2n+1\right ) \pi }{2L}x\right ) dx\right ) \sin \left ( \frac {\left ( 2n+1\right ) \pi }{2L}x\right ) e^{-\left ( \frac {\left ( 2n+1\right ) \pi }{2L}\right ) ^{2}t}\end{align*}
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3.1.1.30 problem number 179
Added June 9, 2019
Solve the heat equation for \(u(x,t)\)
\[ u_t = k u_{xx} \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} u_x(0,t) &= 0 \\ u(L,t) &= T_0 \\ \end{align*}
Where \(T_0>0\) and initial condition is \(u(x,0)=0\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1,0][u][0, t] == 0, u[1, t] == T0};
ic = u[x, 0] == 0;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions ->{T0>0,k>0,L>0}], 60*10]];
sol= sol/.{ K [1]->n};
\[\left \{\left \{u(x,t)\to 2 \underset {n=0}{\overset {\infty }{\sum }}-\frac {2 e^{-\frac {1}{4} k (2 \pi n+\pi )^2 t} \text {T0} \cos (n \pi ) \cos \left (\frac {1}{2} (2 n+1) \pi x\right )}{2 \pi n+\pi }+\text {T0}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k* diff ( u ( x , t ), x$2 ); ic := u(x,0)=0;
bc := D [1]( u )(0, t ) =0,u(L,t)=T0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde,ic,bc],u(x,t)) assuming T0>0,k>0,L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = -\frac {4 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {\cos \left (\frac {\left (1+2 n \right ) \pi x}{2 L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} \left (1+2 n \right )^{2} t}{4 L^{2}}} \left (-1\right )^{n} \operatorname {T0}}{1+2 n}\right )}{\pi }+\operatorname {T0}\]
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3.1.1.31 problem number 180
This is problem 2.3.7, from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ u_t = k u_{xx} \]
Subject to boundary conditions \(u_x(0,t)=0\) , \(u_x(L,t)=0\) with initial conditions \(u(x,0) =f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0, k > 0, t > 0}], 60*10]];
sol = sol /. { K [1] -> n, K [2] -> x};
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}e^{-\frac {k n^2 \pi ^2 t}{L^2}} \cos \left (\frac {n \pi x}{L}\right ) \int _0^L \cos \left (\frac {n \pi x}{L}\right ) f(x) \, dx+\int _0^L f(x) \, dx}{L}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); assume ( L >0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)=f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \frac {\int _{0}^{L}f \left (x \right )d x +2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}f \left (x \right ) \cos \left (\frac {n \pi x}{L}\right )d x \right )}{L}\]
Hand
solution
\begin{align*} \frac {\partial u}{\partial t} & =k\frac {\partial ^{2}u}{\partial x^{2}}\\ u_{x}\left ( 0,t\right ) & =0\\ u_{x}\left ( L,t\right ) & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) \end{align*}
Let \(u\left ( x,t\right ) =T\left ( t\right ) X\left ( x\right ) \) , then the PDE becomes
\[ \frac {1}{k}T^{\prime }X=X^{\prime \prime }T \]
Dividing by \(XT\neq 0\)
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}\]
Since each side depends on different independent variable and both are equal, they must be both equal to same constant, say \(-\lambda \) . Where \(\lambda \) is assumed real.
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}=-\lambda \]
The two ODE’s generated are
\begin{align} T^{\prime }+k\lambda T & =0\tag {1}\\ X^{\prime \prime }+\lambda X & =0\tag {2}\end{align}
Starting with the space ODE equation (2), with corresponding boundary conditions \(\frac {dX}{dx}\left ( 0\right ) =0,\frac {dX}{dx}\left ( L\right ) =0\) . Assuming the solution is \(X\left ( x\right ) =e^{rx}\) , Then the characteristic equation is
\begin{align*} r^{2}+\lambda & =0\\ r^{2} & =-\lambda \\ r & =\pm \sqrt {-\lambda }\end{align*}
The following cases are considered.
case \(\lambda <0\) In this case, \(-\lambda \) and also \(\sqrt {-\lambda }\) are positive. Hence the roots \(\pm \sqrt {-\lambda }\) are both real. Let
\[ \sqrt {-\lambda }=s \]
Where \(s>0\) . This gives the solution
\begin{align*} X\left ( x\right ) & =A\cosh \left ( sx\right ) +B\sinh \left ( sx\right ) \\ \frac {dX}{dx} & =A\sinh \left ( sx\right ) +B\cosh \left ( sx\right ) \end{align*}
Applying the left B.C. gives
\begin{align*} 0 & =\frac {dX}{dx}\left ( 0\right ) \\ & =B\cosh \left ( 0\right ) \\ & =B \end{align*}
The solution becomes \(X\left ( x\right ) =A\cosh \left ( sx\right ) \) and hence\(\frac {dX}{dx}=A\sinh \left ( sx\right ) \) . Applying the right B.C. gives
\begin{align*} 0 & =\frac {dX}{dx}\left ( L\right ) \\ & =A\sinh \left ( sL\right ) \end{align*}
\(A=0\) result in trivial solution. Therefore assuming \(\sinh \left ( sL\right ) =0\) implies \(sL=0\) which is not valid since \(s>0\) and \(L\neq 0\) . Hence only trivial solution results from this case. \(\lambda <0\) is not an eigenvalue
case \(\lambda =0\)
The ODE becomes
\[ \frac {d^{2}X}{dx^{2}}=0 \]
The solution is
\begin{align*} X\left ( x\right ) & =c_{1}x+c_{2}\\ \frac {dX}{dx} & =c_{1}\end{align*}
Applying left boundary conditions gives
\begin{align*} 0 & =\frac {dX}{dx}\left ( 0\right ) \\ & =c_{1}\end{align*}
Hence the solution becomes \(X\left ( x\right ) =c_{2}\) . Therefore \(\frac {dX}{dx}=0\) . Applying the right B.C. provides no information.
Therefore this case leads to the solution \(X\left ( x\right ) =c_{2}\) . Associated with this one eigenvalue, the time equation becomes \(\frac {dT_{0}}{dt}=0\) hence \(T_{0}\) is constant, say \(\alpha \) . Hence the solution \(u_{0}\left ( x,t\right ) \) associated with this \(\lambda =0\) is
\begin{align*} u_{0}\left ( x,t\right ) & =X_{0}T_{0}\\ & =c_{2}\alpha \\ & =A_{0}\end{align*}
where constant \(c_{2}\alpha \) was renamed to \(A_{0}\) to indicate it is associated with \(\lambda =0\) . \(\lambda =0\) is an eigenvalue
case \(\lambda >0\)
Hence \(-\lambda \) is negative, and the roots are both complex.
\[ r=\pm i\sqrt {\lambda }\]
The solution is
\begin{align*} X\left ( x\right ) & =A\cos \left ( \sqrt {\lambda }x\right ) +B\sin \left ( \sqrt {\lambda }x\right ) \\ \frac {dX}{dx} & =-A\sqrt {\lambda }\sin \left ( \sqrt {\lambda }x\right ) +B\sqrt {\lambda }\cos \left ( \sqrt {\lambda }x\right ) \end{align*}
Applying the left B.C. gives
\begin{align*} 0 & =\frac {dX}{dx}\left ( 0\right ) \\ & =B\sqrt {\lambda }\cos \left ( 0\right ) \\ & =B\sqrt {\lambda }\end{align*}
Therefore \(B=0\) as \(\lambda >0\) . The solution becomes \(X\left ( x\right ) =A\cos \left ( \sqrt {\lambda }x\right ) \) and \(\frac {dX}{dx}=-A\sqrt {\lambda }\sin \left ( \sqrt {\lambda }x\right ) \) . Applying the right B.C. gives
\begin{align*} 0 & =\frac {dX}{dx}\left ( L\right ) \\ & =-A\sqrt {\lambda }\sin \left ( \sqrt {\lambda }L\right ) \end{align*}
\(A=0\) gives a trivial solution. Selecting \(\sin \left ( \sqrt {\lambda }L\right ) =0\) gives
\[ \sqrt {\lambda }L=n\pi \qquad n=1,2,3,\cdots \]
Or
\[ \lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \]
Therefore the space solution is
\[ X_{n}\left ( x\right ) =A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \qquad n=1,2,3,\cdots \]
The time solution is found by solving
\[ \frac {dT_{n}}{dt}+k\lambda _{n}T_{n}=0 \]
This has the solution
\begin{align*} T_{n}\left ( t\right ) & =e^{-k\lambda _{n}t}\\ & =e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\qquad n=1,2,3,\cdots \end{align*}
For the same set of eigenvalues. Notice that no need to add a constant here, since it will be absorbed in the \(A_{n}\) when combined in the following step below. Since for \(\lambda =0\) the time solution was found to be constant, and for \(\lambda >0\) the time solution is \(e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\) , then no time solution will grow with time . Time solutions always decay with time as the exponent \(-k\left ( \frac {n\pi }{L}\right ) ^{2}t\) is negative quantity. The solution to the PDE for \(\lambda >0\) is
\[ u_{n}\left ( x,t\right ) =T_{n}\left ( t\right ) X_{n}\left ( x\right ) \qquad n=0,1,2,3,\cdots \]
But for linear system sum of
eigenfunctions is also a solution. Hence
\begin{align*} u\left ( x,t\right ) & =u_{\lambda =0}\left ( x,t\right ) +\sum _{n=1}^{\infty }u_{n}\left ( x,t\right ) \\ & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\end{align*}
From the solution found above, setting \(t=0\) gives
\[ f\left ( x\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \]
Multiplying both sides with \(\cos \left ( \frac {m\pi }{L}x\right ) \) where in this problem \(m=0,1,2,\cdots \) (since there was an eigenvalue associated with \(\lambda =0\) ), and integrating over the domain gives
\begin{align*} \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx & =\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) \left ( A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \right ) dx\\ & =\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\int \cos \left ( \frac {m\pi }{L}x\right ) \sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) dx\\ & =\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\int _{0}^{L}\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {m\pi }{L}x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \end{align*}
Interchanging the order of summation and integration
\begin{equation} \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\sum _{n=1}^{\infty }A_{n}\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx\tag {1}\end{equation}
case \(m=0\)
When \(m=0\) then \(\cos \left ( \frac {m\pi }{L}x\right ) =1\) and the above simplifies to
\[ \int _{0}^{L}f\left ( x\right ) dx=\int _{0}^{L}A_{0}dx+\sum _{n=1}^{\infty }A_{n}\int _{0}^{L}\cos \left ( \frac {n\pi }{L}x\right ) dx \]
But \(\int _{0}^{L}\cos \left ( \frac {n\pi }{L}x\right ) dx=0\) and the above becomes
\begin{align*} \int _{0}^{L}f\left ( x\right ) dx & =\int _{0}^{L}A_{0}dx\\ & =A_{0}L \end{align*}
Therefore
\[ A_{0}=\frac {1}{L}\int _{0}^{L}f\left ( x\right ) dx \]
case \(m>0\)
From (1), one term survives in the integration when only \(n=m\) , hence
\[ \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=A_{0}\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) dx+A_{m}\int _{0}^{L}\cos ^{2}\left ( \frac {m\pi }{L}x\right ) dx \]
But \(\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) dx=0\) and the above becomes
\[ \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=A_{m}\frac {L}{2}\]
Therefore
\[ A_{n}=\frac {2}{L}\int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \]
For \(n=1,2,3,\cdots \)
Therefore the solution is
\begin{align*} u\left ( x,t\right ) & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\\ & =\frac {1}{L}\int _{0}^{L}f\left ( x\right ) dx+\frac {2}{L}\sum _{n=1}^{\infty }\left ( \int _{0}^{L}f\left ( x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx\right ) \cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\end{align*}
In the limit as \(t\rightarrow \infty \) the term \(e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\rightarrow 0\) . What is left is \(A_{0}\) . But \(A_{0}=\frac {1}{L}\int _{0}^{L}f\left ( x\right ) dx\) from above. This quantity is the average of the initial temperature.
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3.1.1.32 problem number 181
Added June 21, 2019
Solve
\[ u_t = k u_{xx} \]
Subject to boundary conditions \(u_x(0,t)=0\) , \(u_x(L,t)=0\) with initial conditions \(u(x,0)=f(x)\) using the following values
\begin{align*} L&=5\\ k&=\frac {1}{100}\\ f(x)&=x \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; k =1/100; L =5; f = x ; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == f;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
sol = sol /. { K [1] -> n, K [2] -> x};
\[\left \{\left \{u(x,t)\to \frac {2}{5} \underset {n=1}{\overset {\infty }{\sum }}\frac {25 \left (-1+(-1)^n\right ) e^{-\frac {n^2 \pi ^2 t}{2500}} \cos \left (\frac {n \pi x}{5}\right )}{n^2 \pi ^2}+\frac {5}{2}\right \}\right \}\]
Maple ✓
restart ; L :=5; k :=1/100; f := x ; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)=f;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \frac {\frac {5 \pi ^{2}}{2}+10 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (\left (-1\right )^{n}-1\right ) \cos \left (\frac {n \pi x}{5}\right ) {\mathrm e}^{-\frac {\pi ^{2} n^{2} t}{2500}}}{n^{2}}\right )}{\pi ^{2}}\]
Hand
solution
The general solution for this type of PDE is given in problem 3.1.1.31 on page 452 as
\begin{equation} u\left ( x,t\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\tag {1}\end{equation}
\(u\left ( x,0\right ) =\) \(f\left ( x\right ) =x,L=5,k=\frac {1}{100}\) . Hence the above becomes
\begin{equation} u\left ( x,t\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{5}x\right ) e^{-\left ( \frac {1}{100}\left ( \frac {n\pi }{5}\right ) ^{2}t\right ) }\tag {2}\end{equation}
\(t=0\) the above becomes
\[ x=A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{5}x\right ) \]
For \(n=0\) orthogonality gives
\begin{align*} \int _{0}^{5}xdx & =\int _{0}^{5}A_{0}dx\\ \frac {25}{2} & =5A_{0}\\ A_{0} & =\frac {5}{2}\end{align*}
For \(n>0\)
\begin{align*} \int _{0}^{5}x\cos \left ( \frac {n\pi }{5}x\right ) dx & =\int _{0}^{5}A_{n}\cos ^{2}\left ( \frac {n\pi }{5}x\right ) dx\\ \frac {25}{n^{2}\pi ^{2}}\left ( -1+\left ( -1\right ) ^{n}\right ) & =\frac {5}{2}A_{n}\\ A_{n} & =\frac {10}{n^{2}\pi ^{2}}\left ( -1+\left ( -1\right ) ^{n}\right ) \end{align*}
Hence the solution (2) becomes
\[ u\left ( x,t\right ) =\frac {5}{2}+\frac {10}{\pi ^{2}}\sum _{n=1}^{\infty }\frac {1}{n^{2}}\left ( -1+\left ( -1\right ) ^{n}\right ) \cos \left ( \frac {n\pi }{5}x\right ) e^{-\frac {n^{2}\pi ^{2}}{2500}t}\]
The following is an animation of the solution
Source code used for the above
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3.1.1.33 problem number 182
This is problem 2.4.1 part(a) from Richard Haberman applied partial differential equations, 5th edition.
Consider the heat equation
\[ u_t = k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u_x(L,t)=0\) . Initial conditions
\(u(x,0) =\left \{ \begin {array} [c]{ccc}0 & & x< \frac {L}{2}\\ 1 & & x> \frac {L}{2} \end {array} \right . \)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == Piecewise [{{0, x < L/2}, {1, x > L/2}}];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0 && k > 0}], 60*10]];
sol = sol /. { K [1] -> n};
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}-\frac {e^{-\frac {k n^2 \pi ^2 t}{L^2}} L \cos \left (\frac {n \pi x}{L}\right ) \sin \left (\frac {n \pi }{2}\right )}{n \pi }}{L}+\frac {1}{2}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)= piecewise (0< x and x<=L/2,0,L/2<x and x<L,1);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming k>0,L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {1}{2}-\frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \sin \left (\frac {n \pi }{2}\right )}{n}\right )}{\pi }\]
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3.1.1.34 problem number 183
Added January 13, 2020
This is problem 4.1.7 Introduction to Partial Differential Equations by Peter Olver ISBN 9783319020983.
Solve
\[ u_t = k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u_x(L,t)=0\) . Initial conditions
\(u(x,0) =\left \{ \begin {array} [c]{ccc}x & & 0<x< \frac {L}{2}\\ 1-x & & \frac {L}{2}<x<L \end {array} \right . Use \) L=1\(. \)
Mathematica ✓
ClearAll [ " Global ` * " ]; L =1; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == Piecewise [{{x, 0<x < L/2}, {1-x, L/2<=x<L}}];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> k > 0], 60*10]];
sol = sol /. K [1] -> n;
\[\left \{\left \{u(x,t)\to 2 \underset {n=1}{\overset {\infty }{\sum }}\frac {4 e^{-k n^2 \pi ^2 t} \cos \left (\frac {n \pi }{2}\right ) \cos (n \pi x) \sin ^2\left (\frac {n \pi }{4}\right )}{n^2 \pi ^2}+\frac {1}{4}\right \}\right \}\]
Maple ✓
restart ; L :=1; pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)= piecewise (0< x and x<=L/2,x,L/2<x and x<L,1-x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {1}{4}+\frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {\cos \left (n \pi x \right ) {\mathrm e}^{-k \,\pi ^{2} n^{2} t} \left (-2 \cos \left (\frac {n \pi }{2}\right )+1+\left (-1\right )^{n}\right )}{n^{2}}\right )\right )}{\pi ^{2}}\]
Hand
solution
Let \(u\left ( x,t\right ) =T\left ( t\right ) X\left ( x\right ) \) , then the PDE becomes
\[ \frac {1}{k}T^{\prime }X=X^{\prime \prime }T \]
Dividing by \(XT\neq 0\)
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}\]
Since each side depends on different independent variable and both are equal, they must be both equal to same constant, say \(-\lambda \) . Where \(\lambda \) is assumed real.
\[ \frac {1}{k}\frac {T^{\prime }}{T}=\frac {X^{\prime \prime }}{X}=-\lambda \]
The two ODE’s generated are
\begin{equation} T^{\prime }+k\lambda T=0\tag {1}\end{equation}
\begin{align} X^{\prime \prime }+\lambda X & =0\tag {2}\\ X^{\prime }\left ( 0\right ) & =0\nonumber \\ X^{\prime }\left ( L\right ) & =0\nonumber \end{align}
Starting with the eigenvalue ODE equation (2). The following cases are considered.
case \(\lambda <0\)
In this case, \(-\lambda \) is positive. Let \(-\lambda =\omega ^{2}\) . Hence the ODE is \(X^{\prime \prime }-\omega ^{2}X=0\) and the solution becomes
\[ X\left ( x\right ) =C_{1}\cosh \left ( \omega x\right ) +C_{2}\sinh \left ( \omega x\right ) \]
Therefore
\[ X^{\prime }=C_{1}\sinh \left ( \omega x\right ) +C_{2}\cosh \left ( \omega x\right ) \]
Applying the left B.C. gives
\[ 0=C_{2}\]
Therefore the solution becomes \(X\left ( x\right ) =C_{1}\cosh \left ( \omega x\right ) \) and \(X^{\prime }\left ( x\right ) =C_{1}\sinh \left ( \omega x\right ) \) . Applying the right B.C. gives
\[ 0=C_{1}\sinh \left ( \omega L\right ) \]
For non-trivial solution we want \(\sinh \left ( \omega L\right ) =0\) . But this is not possible since \(\sinh \) is zero when its argument is zero, which is not the case here. Hence only trivial solution results from this case. \(\lambda <0\) is not an
eigenvalue.
case \(\lambda =0\)
The solution is
\begin{align*} X\left ( x\right ) & =c_{1}x+c_{2}\\ X^{\prime }\left ( x\right ) & =c_{1}\end{align*}
Applying left boundary conditions gives
\[ 0=c_{1}\]
Hence the solution becomes \(X\left ( x\right ) =c_{2}\) . Therefore \(\frac {dX}{dx}=0\) . Applying the right B.C. provides no information. Any \(c_{2}\) will work. Therefore this case leads to the solution \(X\left ( x\right ) =c_{2}\) . Associated with this one eigenvalue, the time equation becomes \(T_{0}^{\prime }\left ( t\right ) =0\) hence \(T_{0}\left ( t\right ) \) is a constant. Hence the solution \(u_{0}\left ( x,t\right ) \) associated with this \(\lambda =0\) is
\begin{align*} u_{0}\left ( x,t\right ) & =X_{0}T_{0}\\ & =A_{0}\end{align*}
where constant \(c_{2}T_{0}\) was renamed to \(\frac {A_{0}}{2}\) to indicate it is associated with \(\lambda =0\) . \(\lambda =0\) is an eigenvalue\(\frac {A_{0}}{2}\) .
case \(\lambda >0\)
The solution is
\begin{align*} X\left ( x\right ) & =c_{1}\cos \left ( \sqrt {\lambda }x\right ) +c_{2}\sin \left ( \sqrt {\lambda }x\right ) \\ X^{\prime }\left ( x\right ) & =-c_{1}\sqrt {\lambda }\sin \left ( \sqrt {\lambda }x\right ) +c_{2}\sqrt {\lambda }\cos \left ( \sqrt {\lambda }x\right ) \end{align*}
Applying the left B.C. gives
\[ 0=c_{2}\sqrt {\lambda }\]
Therefore \(c_{2}=0\) as \(\lambda >0\) . The solution becomes
\[ X\left ( x\right ) =c_{1}\cos \left ( \sqrt {\lambda }x\right ) \]
And \(X^{\prime }\left ( x\right ) =-c_{1}\sqrt {\lambda }\sin \left ( \sqrt {\lambda }x\right ) \) . Applying the right B.C. gives
\[ 0=-c_{1}\sqrt {\lambda }\sin \left ( \sqrt {\lambda }L\right ) \]
\(c_{1}=0\) gives a trivial solution. Selecting \(\sin \left ( \sqrt {\lambda }L\right ) =0\) gives
\[ \sqrt {\lambda }L=n\pi \qquad n=1,2,3,\cdots \]
Or
\[ \lambda _{n}=\left ( \frac {n\pi }{L}\right ) ^{2}\qquad n=1,2,3,\cdots \]
Therefore the eigenfunctions are
\[ X_{n}\left ( x\right ) =\cos \left ( \frac {n\pi }{L}x\right ) \qquad n=1,2,3,\cdots \]
The time solution is found by solving
\[ T_{n}^{\prime }\left ( t\right ) +k\lambda _{n}T_{n}\left ( t\right ) =0 \]
This has the solution
\begin{align*} T_{n}\left ( t\right ) & =A_{n}e^{-k\lambda _{n}t}\\ & =A_{n}e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\qquad n=1,2,3,\cdots \end{align*}
The solution to the PDE is
\[ u_{n}\left ( x,t\right ) =T_{n}\left ( t\right ) X_{n}\left ( x\right ) \qquad n=0,1,2,3,\cdots \]
But for linear system sum of eigenfunctions is also a solution. Hence
\begin{align} u\left ( x,t\right ) & =u_{0}\left ( x,t\right ) +\sum _{n=1}^{\infty }u_{n}\left ( x,t\right ) \nonumber \\ & =\frac {A_{0}}{2}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\tag {1}\end{align}
From the solution found above, setting \(t=0\) gives
\[ f\left ( x\right ) =\frac {A_{0}}{2}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \]
Hence \(A_{0},A_{n}\) are the Fourier \(\cos \) coefficients for the function \(f\left ( x\right ) \) . Doing an even extension of \(f\left ( x\right ) \) from \(\left [ -L,L\right ] \) , then \(\frac {A_{0}}{2}\) is the average of the function \(f\left ( x\right ) \) over \(\left [ -L,L\right ] \) . But this average is seen as \(\frac {2\left ( \frac {1}{2}\times \frac {1}{2}\right ) }{2}=\frac {1}{4}\) . The term \(\frac {1}{2}\times \frac {1}{2}\) is the area of \(f\left ( x\right ) \) from \(\left [ 0,L\right ] \) .
\[ \frac {A_{0}}{2}=\frac {1}{4}\]
For \(A_{n}\)
\[ A_{n}=\frac {1}{L}\int _{-L}^{L}f\left ( x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \]
Replacing \(L=1\) and using the definition of \(f\left ( x\right ) \) given above gives
\[ A_{n}=\int _{-1}^{1}f\left ( x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \]
But \(f\left ( x\right ) \) is even (after even extending) and \(\cos \) is even, hence the above
becomes
\begin{align} A_{n} & =2\int _{0}^{1}f\left ( x\right ) \cos \left ( n\pi x\right ) dx\nonumber \\ & =2\left ( \int _{0}^{\frac {1}{2}}x\cos \left ( n\pi x\right ) dx+\int _{\frac {1}{2}}^{1}\left ( 1-x\right ) \cos \left ( n\pi x\right ) dx\right ) \nonumber \\ & =2\left ( \int _{0}^{\frac {1}{2}}x\cos \left ( n\pi x\right ) dx+\int _{\frac {1}{2}}^{1}\cos \left ( n\pi x\right ) dx-\int _{\frac {1}{2}}^{1}x\cos \left ( n\pi x\right ) dx\right ) \tag {2}\end{align}
But
\begin{align} \int _{a}^{b}x\cos \left ( n\pi x\right ) dx & =\frac {1}{n\pi }\left [ x\sin \left ( n\pi x\right ) \right ] _{a}^{b}-\frac {1}{n\pi }\int _{a}^{b}\sin \left ( n\pi x\right ) dx\nonumber \\ & =\frac {1}{n\pi }\left [ x\sin \left ( n\pi x\right ) \right ] _{a}^{b}+\frac {1}{n^{2}\pi ^{2}}\left [ \cos \left ( n\pi x\right ) \right ] _{a}^{b}\tag {3}\end{align}
When \(a=0,b=\frac {1}{2}\) the above gives
\begin{align} \int _{0}^{\frac {1}{2}}x\cos \left ( n\pi x\right ) dx & =\frac {1}{n\pi }\left [ x\sin \left ( n\pi x\right ) \right ] _{0}^{\frac {1}{2}}+\frac {1}{n^{2}\pi ^{2}}\left [ \cos \left ( n\pi x\right ) \right ] _{0}^{\frac {1}{2}}\nonumber \\ & =\frac {1}{n\pi }\left ( \frac {1}{2}\sin \left ( \frac {n\pi }{2}\right ) \right ) +\frac {1}{n^{2}\pi ^{2}}\left ( \cos \left ( \frac {n\pi }{2}\right ) -1\right ) \nonumber \\ & =\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\left ( \cos \left ( \frac {n\pi }{2}\right ) -1\right ) \nonumber \\ & =\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\tag {4}\end{align}
And when \(a=\frac {1}{2},b=1\) (3) gives
\begin{align} \int _{\frac {1}{2}}^{1}x\cos \left ( n\pi x\right ) dx & =\frac {1}{n\pi }\left [ x\sin \left ( n\pi x\right ) \right ] _{\frac {1}{2}}^{1}+\frac {1}{n^{2}\pi ^{2}}\left [ \cos \left ( n\pi x\right ) \right ] _{\frac {1}{2}}^{1}\nonumber \\ & =\frac {1}{n\pi }\left [ \sin \left ( n\pi \right ) -\frac {1}{2}\sin \left ( \frac {n\pi }{2}\right ) \right ] +\frac {1}{n^{2}\pi ^{2}}\left [ \cos \left ( n\pi \right ) -\cos \left ( \frac {n\pi }{2}\right ) \right ] \nonumber \\ & =-\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( n\pi \right ) -\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) \tag {5}\end{align}
Substituting (4,5) into (2) gives
\begin{align*} \frac {A_{n}}{2} & =\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\\ & +\int _{\frac {1}{2}}^{1}\cos \left ( n\pi x\right ) dx\\ & -\left ( -\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( n\pi \right ) -\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) \right ) \end{align*}
Or
\begin{align*} \frac {A_{n}}{2} & =\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\\ & +\frac {1}{n\pi }\overset {0}{\overbrace {\sin \left ( n\pi \right ) }}-\frac {1}{n\pi }\sin \left ( \frac {n\pi }{2}\right ) \\ & +\frac {1}{2n\pi }\sin \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\cos \left ( n\pi \right ) +\frac {1}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) \end{align*}
Or
\begin{align*} \frac {A_{n}}{2} & =\left ( \frac {1}{n\pi }\sin \left ( \frac {n\pi }{2}\right ) +\frac {2}{n^{2}\pi ^{2}}\cos \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\right ) -\frac {1}{n\pi }\sin \left ( \frac {n\pi }{2}\right ) -\frac {1}{n^{2}\pi ^{2}}\cos \left ( n\pi \right ) \\ & =\frac {2\cos \left ( \frac {n\pi }{2}\right ) -1-\left ( -1\right ) ^{n}}{n^{2}\pi ^{2}}\end{align*}
Therefore the solution (1) becomes, after replacing \(L=1\)
\begin{equation} u\left ( x,t\right ) =\frac {1}{4}+2\sum _{n=1}^{\infty }\frac {2\cos \left ( \frac {n\pi }{2}\right ) -1-\left ( -1\right ) ^{n}}{n^{2}\pi ^{2}}\cos \left ( n\pi x\right ) e^{-kn^{2}\pi ^{2}t}\nonumber \end{equation}
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3.1.1.35 problem number 184
This is problem 2.4.1 part(b) from Richard Haberman applied partial differential equations, 5th edition.
Solve the heat equation
\[ u_t= k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u_x(L,t)=0\) with the temperature initially \(u(x,0)=6+ 4 \cos \left ( \frac {3 \pi x}{L} \right )\)
Mathematica ✓
ClearAll [ " Global ` * " ]; NumericQ [ L ] =. ;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == 6 + 4* Cos [(3* Pi *x)/L];
NumericQ [ L ] = True ;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
sol = sol /. { K [1] -> n};
NumericQ [L] =. ;
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}\frac {2 e^{-\frac {k n^2 \pi ^2 t}{L^2}} L \left (n^2-27\right ) \cos \left (\frac {n \pi x}{L}\right ) \sin (n \pi )}{n \left (n^2-9\right ) \pi }}{L}+6\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); assume ( L >0 and k>0);
pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)=6+4* cos (3* Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = 6+4 \cos \left (\frac {3 \pi x}{L}\right ) {\mathrm e}^{-\frac {9 k \,\pi ^{2} t}{L^{2}}}\]
Hand
solution
The general solution for this type of PDE is given in problem 3.1.1.31 on page 452 as
\begin{equation} u\left ( x,t\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\tag {1}\end{equation}
\(u\left ( x,0\right ) =\) \(f\left ( x\right ) =6+4\cos \frac {3\pi x}{L}\) . Hence at \(t=0\) the above becomes
\begin{align*} f\left ( x\right ) & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \\ 6+4\cos \frac {3\pi x}{L} & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \end{align*}
Comparing terms shows that
\begin{align*} A_{0} & =6\\ A_{3} & =4 \end{align*}
And all other \(A_{n}=0\) . Hence the solution (1) is
\[ u\left ( x,t\right ) =6+4\cos \left ( \frac {3\pi }{L}x\right ) e^{-k\left ( \frac {3\pi }{L}\right ) ^{2}t}\]
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3.1.1.36 problem number 185
This is problem 2.4.1 part(c) from Richard Haberman applied partial differential equations, 5th edition.
Solve the heat equation
\[ u_t = k u_{xx} \]
The boundary conditions are \(u_x(0,t)=0\) , \(u_x(L,t)=0\) with the temperature initially \(u(x,0)=-2 \sin \frac {\pi x}{L}\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == -2* Sin [( Pi *x)/L];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {L > 0 && k > 0}], 60*10]];
sol = sol /. K [1] -> n;
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}\frac {2 \left (1+(-1)^n\right ) e^{-\frac {k n^2 \pi ^2 t}{L^2}} L \cos \left (\frac {n \pi x}{L}\right )}{\left (n^2-1\right ) \pi }}{L}-\frac {4}{\pi }\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); assume ( L >0 and k>0);
pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; ic := u(x,0)=-2* sin ( Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = \frac {-4+4 \left (\moverset {\infty }{\munderset {n =2}{\sum }}\frac {\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \left (1+\left (-1\right )^{n}\right )}{n^{2}-1}\right )}{\pi }\]
Hand
solution
The general solution for this type of PDE is given in problem 3.1.1.31 on page 452 as
\begin{equation} u\left ( x,t\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\tag {1}\end{equation}
\(t=0\) the above becomes
\begin{align*} f\left ( x\right ) & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \\ -2\sin \frac {\pi x}{L} & =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \end{align*}
Multiplying both sides by \(\cos \left ( \frac {m\pi }{L}x\right ) \) and integrating gives
\begin{align*} -2\int _{0}^{L}\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx & =\int _{0}^{L}\left ( A_{0}\cos \left ( \frac {m\pi }{L}x\right ) +\cos \left ( \frac {m\pi }{L}x\right ) \sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {n\pi }{L}x\right ) \right ) dx\\ & =\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\int _{0}^{L}\sum _{n=1}^{\infty }A_{n}\cos \left ( \frac {m\pi }{L}x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \end{align*}
Interchanging the order of integration and summation
\[ \int _{0}^{L}-2\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\sum _{n=1}^{\infty }A_{n}\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \]
Case \(m=0\)
The above becomes
\[ -2\int _{0}^{L}\sin \left ( \frac {\pi x}{L}\right ) dx=\int _{0}^{L}A_{0}dx+\sum _{n=1}^{\infty }A_{n}\int _{0}^{L}\cos \left ( \frac {n\pi }{L}x\right ) dx \]
But \(\int _{0}^{L}\cos \left ( \frac {n\pi }{L}x\right ) dx=0\) hence
\begin{align*} \int _{0}^{L}-2\sin \left ( \frac {\pi x}{L}\right ) dx & =\int _{0}^{L}A_{0}dx\\ A_{0}L & =-2\int _{0}^{L}\sin \left ( \frac {\pi x}{L}\right ) dx\\ A_{0}L & =-2\left ( -\frac {\cos \left ( \frac {\pi x}{L}\right ) }{\frac {\pi }{L}}\right ) _{0}^{L}\\ & =-\frac {2L}{\pi }\left ( -\cos \left ( \frac {\pi L}{L}\right ) +\cos \left ( \frac {\pi 0}{L}\right ) \right ) \\ & =-\frac {2L}{\pi }\left ( -\left ( -1\right ) +1\right ) \\ & =-\frac {4L}{\pi }\end{align*}
Hence
\[ A_{0}=\frac {-4}{\pi }\]
Case \(m>0\)
\[ \int _{0}^{L}-2\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=\int _{0}^{L}A_{0}\cos \left ( \frac {m\pi }{L}x\right ) dx+\sum _{n=1}^{\infty }A_{n}\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \]
One term survives the summation resulting in
\[ \int _{0}^{L}-2\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx=\frac {-4}{\pi }\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) dx+A_{m}\int _{0}^{L}\cos ^{2}\left ( \frac {m\pi }{L}x\right ) dx \]
But \(\int _{0}^{L}\cos \left ( \frac {m\pi }{L}x\right ) dx=0\) and \(\int _{0}^{L}\cos ^{2}\left ( \frac {m\pi }{L}x\right ) dx=\frac {L}{2}\) , therefore
\begin{align*} \int _{0}^{L}-2\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {m\pi }{L}x\right ) dx & =A_{m}\frac {L}{2}\\ A_{n} & =\frac {-4}{L}\int _{0}^{L}\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx \end{align*}
But
\[ \int _{0}^{L}\sin \left ( \frac {\pi x}{L}\right ) \cos \left ( \frac {n\pi }{L}x\right ) dx=\frac {-L\left ( 1+\cos \left ( n\pi \right ) \right ) }{\pi \left ( n^{2}-1\right ) }\]
Therefore
\begin{align*} A_{n} & =4\frac {\left ( 1+\cos \left ( n\pi \right ) \right ) }{\pi \left ( n^{2}-1\right ) }\\ & =4\frac {\left ( -1\right ) ^{n}+1}{\pi \left ( n^{2}-1\right ) }\qquad n=1,2,3,\cdots \end{align*}
Hence the solution becomes
\[ u\left ( x,t\right ) =\frac {-4}{\pi }+\frac {4}{\pi }\sum _{n=1}^{\infty }\frac {\left ( -1\right ) ^{n}+1}{\left ( n^{2}-1\right ) }\cos \left ( \frac {n\pi }{L}x\right ) e^{-k\left ( \frac {n\pi }{L}\right ) ^{2}t}\]
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3.1.1.37 problem number 186
This is problem 2.4.1 part(d) from Richard Haberman applied partial differential equations, 5th edition.
Solve the heat equation
\[ \frac { \partial u}{\partial t} = k \frac { \partial ^2 u}{\partial x^2} \]
The boundary conditions are \(\frac {\partial u}{\partial x}(0,t)=0\) and \(\frac {\partial u}{\partial x}(L,t)=0\) with the temperature initially \(u(x,0)=-3 \cos \frac {8 \pi x}{L}\)
Mathematica ✓
ClearAll [ " Global ` * " ]; NumericQ [ L ] =. ;
pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == -3* Cos [(8* Pi *x)/L];
NumericQ [ L ] = True ;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
sol = sol /. { K [1] -> n};
NumericQ [L] =. ;
\[\left \{\left \{u(x,t)\to \frac {2 \underset {n=1}{\overset {\infty }{\sum }}-\frac {3 e^{-\frac {k n^2 \pi ^2 t}{L^2}} L n \cos \left (\frac {n \pi x}{L}\right ) \sin (n \pi )}{\left (n^2-64\right ) \pi }}{L}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; assume ( L >0 and k>0);
ic := u(x,0)=-3* cos (8* Pi * x / L ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t ))), output = ' realtime ' )); \[u \left (x , t\right ) = -3 \cos \left (\frac {8 \pi x}{L}\right ) {\mathrm e}^{-\frac {64 k \,\pi ^{2} t}{L^{2}}}\]
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3.1.1.38 problem number 187
Added December 20, 2018.
Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018
Solve the heat equation for \(u(x,t)\)
\[ u_t= 13 u_{xx} \]
For \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &= 0\\ \frac {\partial u}{\partial x}(1,t) &= 1 \end{align*}
Initial condition is \(u(x,0)=\frac {1}{2} x^2+x\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == 13* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][1, t] == 1};
ic = u[x, 0] == (1*x^2)/2 + x;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t], 60*10]];
\[\left \{\left \{u(x,t)\to \frac {1}{2} \left (2 \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 \left (-1+(-1)^{K[1]}\right ) e^{-13 \pi ^2 t K[1]^2} \cos (\pi x K[1])}{\pi ^2 K[1]^2}+26 t+x^2+1\right )\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t), t) = 13*( diff ( u ( x , t), x$2));
bc := eval ( diff ( u ( x , t ), x ), x =0)=0 , eval ( diff ( u ( x , t ), x ), x =1)=1; ic := u(x,0)=1/2*x^2+x;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, bc,ic],u(x,t))),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {4 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\cos \left (n \pi x \right ) {\mathrm e}^{-13 \pi ^{2} n^{2} t} \left (-1+\left (-1\right )^{n}\right )}{n^{2}}\right )+\left (x^{2}+26 t +1\right ) \pi ^{2}}{2 \pi ^{2}}\]
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3.1.1.39 problem number 188
This problem is taken from Maple primes post
Left end insulated, right end has convection heat loss https://www.mapleprimes.com/posts/209681-Solving-PDEs-With-Initial-And-Boundary
Solve the heat equation
\[ u_t = k u_{xx} \]
The boundary conditions are, on the left end \(\frac {\partial u}{\partial x}(0,t)=0\) and on the right end \(\frac {\partial u}{\partial x}(1,t)=-u(1,t)\) with the temperature initially \(u(x,0)=1-\frac {1}{4} x^3\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
ic = u[x, 0] == 1 - (1*x^3)/4;
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][1, t] == -u[1, t]};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {k > 0, t > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \underset {K[1]=1}{\overset {\infty }{\sum }}-\frac {e^{-k t K[2,K[1]]} \cos \left (x \sqrt {K[2,K[1]]}\right ) \left (3 \cos \left (\sqrt {K[2,K[1]]}\right ) (K[2,K[1]]-2)-3 \sqrt {K[2,K[1]]} (K[2,K[1]]+2) \sin \left (\sqrt {K[2,K[1]]}\right )+6\right )}{\sqrt {2} \sqrt {3-\cos \left (2 \sqrt {K[2,K[1]]}\right )} K[2,K[1]]^2 \sqrt {\sin ^2\left (\sqrt {K[2,K[1]]}\right )+1}} & \tan \left (\sqrt {K[2,K[1]]}\right )=\frac {1}{\sqrt {K[2,K[1]]}}\land K[1]\in \mathbb {Z}\land K[1]\geq 1\land K[2,K[1]]>0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t) = k*( diff ( u ( x , t ), x, x));
ic := u(x,0) = 1-(1/4)*x^3;
bc := eval ( diff ( u ( x , t ), x), x = 0) = 0, eval ( diff ( u ( x , t ), x), x = 1)+u(1,t) = 0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming t>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {3 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {{\mathrm e}^{-k \lambda _{n}^{2} t} \left (\left (-\lambda _{n}^{2}+2\right ) \cos \left (\lambda _{n}\right )-2+\left (\lambda _{n}^{3}+2 \lambda _{n}\right ) \sin \left (\lambda _{n}\right )\right ) \cos \left (\lambda _{n} x \right )}{\lambda _{n}^{3} \left (\sin \left (\lambda _{n}\right ) \cos \left (\lambda _{n}\right )+\lambda _{n}\right )}\right )}{2}\boldsymbol {\operatorname {where}}\left \{\tan \left (\lambda _{n}\right ) \lambda _{n}-1=0\wedge 0<\lambda _{n}\right \}\]
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3.1.1.40 problem number 189
Added July 2, 2018. Taken from Maple 2018.1 document, originally exercise 6.25 from Pinchover and Rubinstein.
Solve the heat equation
\[ u_t = k u_{xx} + \cos (w t) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &= 0 \\ \frac {\partial u}{\partial x}(L,t) &= 0 \\ \end{align*}
Initial condition is \(u(x,0)=x\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Cos [w*t];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == x;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t, Assumptions -> {L > 0, t > 0, k > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 \left (-1+(-1)^{K[1]}\right ) e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} L \cos \left (\frac {\pi x K[1]}{L}\right )}{\pi ^2 K[1]^2}+\frac {L}{2}+\frac {\sin (t w)}{w}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t), t) = k*( diff ( u ( x , t), x, x))+ cos ( w * t ); bc := ( D [1]( u ))( L , t) = 0, ( D [1]( u ))(0, t) = 0;
ic := u(x, 0) = x;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde,ic,bc],u(x,t)) assuming L>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {L}{2}+\frac {2 L \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \left (\left (-1\right )^{n}-1\right )}{n^{2}}\right )}{\pi ^{2}}+\frac {\sin \left (w t \right )}{w}\]
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3.1.1.41 problem number 190
Added March 18, 2018.
Solve the heat equation
\[ \frac { \partial u}{\partial t}= k \frac { \partial ^2 u}{\partial x^2} + \left ( e^{-c t} \sin \left ( \frac {2 \pi x}{L} \right ) \right ) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &= 0 \\ \frac {\partial u}{\partial x}(L,t) &= 0 \\ \end{align*}
Initial condition is \(u(x,0)=f(x)\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Exp [-(c*t)]* Sin [(2* Pi *x)/L];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t, Assumptions -> {L > 0, k > 0, t > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} \cos \left (\frac {\pi x K[1]}{L}\right ) \left (e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} \int _0^L \frac {\sqrt {2} \cos \left (\frac {\pi x K[1]}{L}\right ) f(x)}{\sqrt {L}} \, dx+\left (\begin {array}{cc} \{ & \begin {array}{cc} \frac {2 \sqrt {2} \left (-1+(-1)^{K[1]}\right ) e^{-t \left (\frac {k \pi ^2 K[1]^2}{L^2}+c\right )} \left (-e^{c t}+e^{\frac {k \pi ^2 t K[1]^2}{L^2}}\right ) L^{5/2}}{\pi \left (K[1]^2-4\right ) \left (k \pi ^2 K[1]^2-c L^2\right )} & K[1]\neq 2 \\ 0 & \text {True} \\\end {array} \\\end {array}\right )\right )}{\sqrt {L}}+\frac {\int _0^L \frac {f(x)}{\sqrt {L}} \, dx}{\sqrt {L}}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )+( exp (- c * t )* sin (2* Pi * x / L )); ic := u(x,0)=f(x);
bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde,ic,bc],u(x,t)) assuming L>0,t>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {\int _{0}^{L}f \left (x \right )d x}{L}+2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}f \left (x \right ) \cos \left (\frac {n \pi x}{L}\right )d x}{L}\right )-\frac {8 L^{2} \left (\frac {3 \left (\moverset {\infty }{\munderset {n =3}{\sum }}\left (-\frac {\cos \left (\frac {n \pi x}{L}\right ) \left (\left (-1\right )^{n}-1\right ) \left (L^{2} c -\pi ^{2} k \right ) \left ({\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}}-{\mathrm e}^{-c t}\right )}{\left (n^{2}-4\right ) \left (-\pi ^{2} k \,n^{2}+L^{2} c \right )}\right )\right )}{2}+\cos \left (\frac {\pi x}{L}\right ) \left ({\mathrm e}^{-c t}-{\mathrm e}^{-\frac {k \,\pi ^{2} t}{L^{2}}}\right )\right )}{3 \pi \left (L^{2} c -\pi ^{2} k \right )}\]
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3.1.1.42 problem number 191
Added June 9, 2019
Consider the heat equation
\[ u_t = k u_{xx} - \beta u \]
Suppose the boundary conditions are \(u_x(0,t)=0,u_x(\pi ,t)=0\) , solve with the temperature initially \(u\left ( x,0\right ) =x\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] - beta*u[x, t];
bc = { Derivative [1,0][u][0, t] == 0, Derivative [1,0][u][ Pi , t] == 0};
ic = u[x, 0] == x;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {beta > 0,k>0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 \left (-1+(-1)^{K[1]}\right ) e^{-t \left (k K[1]^2+\beta \right )} \cos (x K[1])}{\pi K[1]^2}+\frac {1}{2} \pi e^{-\beta t}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )- beta * u ( x , t ); bc := D [1]( u )(0, t )=0, D [1]( u )( Pi , t )=0; ic := u(x,0)=x;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming beta>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {\pi \,{\mathrm e}^{-\beta t}}{2}+\frac {2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (\left (-1\right )^{n}-1\right ) \cos \left (n x \right ) {\mathrm e}^{-t \left (k \,n^{2}+\beta \right )}}{n^{2}}\right )}{\pi }\]
Hand
solution
Solve
\[ \frac {\partial u}{\partial t}=k\frac {\partial ^{2}u}{\partial x^{2}}-\beta u\qquad 0<x<\pi ,t>0 \]
with, \(k>0,\beta >0\)
\begin{align*} \frac {\partial u\left ( 0,t\right ) }{\partial x} & =0\\ \frac {\partial u\left ( \pi ,t\right ) }{\partial x} & =0 \end{align*}
And initial conditions
\[ u\left ( x,0\right ) =x \]
Solution
Let \(u=X\left ( x\right ) T\left ( t\right ) \) . Substituting into the PDE gives
\[ T^{\prime }X=kX^{\prime \prime }T-\beta XT \]
Dividing by \(XT\neq 0\) gives
\begin{align*} \frac {T^{\prime }}{T} & =\frac {kX^{\prime \prime }}{X}-\beta \\ \frac {T^{\prime }}{T}+\beta & =\frac {kX^{\prime \prime }}{X}\\ \frac {T^{\prime }}{kT}+\frac {\beta }{k} & =\frac {X^{\prime \prime }}{X}\end{align*}
Since each sides depends on different variable and both are equal, they must be equal to same constant, say \(-\lambda \)
\[ \frac {T^{\prime }}{\alpha T}+\frac {\beta }{\alpha }=\frac {X^{\prime \prime }}{X}=-\lambda \]
This gives two ODE’s to solve
\begin{align} X^{\prime \prime }+\lambda X & =0\tag {1}\\ X^{\prime }\left ( 0\right ) & =0\nonumber \\ X^{\prime }\left ( \pi \right ) & =0\nonumber \end{align}
And
\begin{align} \frac {T^{\prime }}{kT}+\frac {\beta }{k} & =-\lambda \tag {2}\\ T^{\prime }+\beta T & =-\lambda kT\nonumber \\ T^{\prime }+T\left ( \lambda k+\beta \right ) & =0\nonumber \end{align}
Starting with (1).
Assuming \(\lambda <0\) , the solution is
\begin{align*} X\left ( x\right ) & =A\cosh \left ( \sqrt {-\lambda }x\right ) +B\sinh \left ( \sqrt {-\lambda }x\right ) \\ X^{\prime } & =\sqrt {-\lambda }A\sinh \left ( \sqrt {-\lambda }x\right ) +\sqrt {-\lambda }B\cosh \left ( \sqrt {-\lambda }x\right ) \end{align*}
Applying first B.C. gives \(0=\sqrt {-\lambda }B\) , hence \(B=0\) . Therefore the solution becomes \(X\left ( x\right ) =A\cosh \left ( \sqrt {-\lambda }x\right ) \) and \(X^{\prime }=\sqrt {-\lambda }A\sinh \left ( \sqrt {-\lambda }x\right ) \) . Applying second B.C. gives
\[ 0=\sqrt {-\lambda }A\sinh \left ( \sqrt {-\lambda }\pi \right ) \]
But \(\sinh \left ( \sqrt {-\lambda }\pi \right ) =0\) only when its argument is zero, which is not the case here. This means \(A=0\) , leading to trivial solution. Therefore \(\lambda <0\) is not eigenvalue.
Assuming \(\lambda =0\) . The solution is \(X=Ax+B\) . Hence \(X^{\prime }=A\) . Applying first B.C. Gives \(A=0\) . Hence solution is \(X=B\) . Second boundary condition gives no additional information. Therefore \(X=1\) (constant) is the eigenfunction associated with \(\lambda =0\) .
Assuming \(\lambda >0\) . The solution is
\begin{align*} X & =A\cos \left ( \sqrt {\lambda }x\right ) +B\sin \left ( \sqrt {\lambda }x\right ) \\ X^{\prime } & =-\sqrt {\lambda }A\sin \left ( \sqrt {\lambda }x\right ) +\sqrt {\lambda }B\cos \left ( \sqrt {\lambda }x\right ) \end{align*}
Applying first B.C. gives \(0=\sqrt {\lambda }B\) , hence \(B=0\) . and the solution becomes \(X=A\cos \left ( \sqrt {\lambda }x\right ) ,X^{\prime }=-\sqrt {\lambda }A\sin \left ( \sqrt {\lambda }x\right ) \) . Applying second B.C. gives
\[ 0=-\sqrt {\lambda }A\sin \left ( \sqrt {\lambda }\pi \right ) \]
Therefore
\begin{align*} \sin \left ( \sqrt {\lambda }\pi \right ) & =0\\ \sqrt {\lambda }\pi & =n\pi \qquad n=1,2,3,\cdots \\ \lambda & =n^{2}\qquad n=1,2,3,\cdots \end{align*}
Therefore the space solution is
\begin{equation} X\left ( x\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( nx\right ) \tag {3}\end{equation}
\[ T^{\prime }+T\left ( \lambda k+\beta \right ) =0 \]
Integrating factor is \(e^{\int \lambda k+\beta dt}=e^{\left ( \lambda k+\beta \right ) t}\) . Hence \(\frac {d}{dt}\left ( Te^{\left ( \lambda k+\beta \right ) t}\right ) =0\) or \(Te^{\left ( \lambda k+\beta \right ) t}=c\) where \(c\) is constant. Therefore
\[ T\left ( t\right ) =ce^{-\left ( \lambda k+\beta \right ) t}\]
For \(\lambda =0\) , the solution is
\[ T_{0}\left ( t\right ) =c_{0}e^{-\beta t}\]
and for \(\lambda >0\) , the solution is
\[ T_{n}\left ( t\right ) =c_{n}e^{-\left ( n^{2}k+\beta \right ) t}\]
Hence the time domain solution is
\begin{equation} T\left ( t\right ) =c_{0}e^{-\beta t}+\sum _{n=1}^{\infty }c_{n}e^{-\left ( n^{2}k+\beta \right ) t}\tag {4}\end{equation}
\begin{align} u_{n} & =X_{n}T_{n}\nonumber \\ u\left ( x,t\right ) & =\sum _{n=0}^{\infty }X_{n}T_{n}\nonumber \\ & =c_{0}e^{-\beta t}+\sum _{n=1}^{\infty }b_{n}\cos \left ( nx\right ) e^{-\left ( n^{2}k+\beta \right ) t}\tag {5}\end{align}
where both constants from space and time eigenfunctions are combined into one constant \(b_{n}\) above in the sum. Now initial conditions are used to find coefficients. At \(t=0\)
\begin{align*} u\left ( x,0\right ) & =c_{0}+\sum _{n=1}^{\infty }b_{n}\cos \left ( nx\right ) \\ x & =c_{0}+\sum _{n=1}^{\infty }b_{n}\cos \left ( nx\right ) \end{align*}
Multiplying both sides by \(\cos \left ( mx\right ) \) and integrating gives
\[ \int _{0}^{\pi }x\cos mxdx=\int _{0}^{\pi }\left ( c_{0}\cos mx+\sum _{n=1}^{\infty }b_{n}\cos mx\cos \left ( nx\right ) \right ) dx \]
For \(m=0\) , all terms in the sum in RHS vanish, and only \(c_{0}\) left
\begin{align*} \int _{0}^{\pi }xdx & =\int _{0}^{\pi }c_{0}dx\\ & =c_{0}\pi \end{align*}
Hence
\begin{align} c_{0} & =\frac {1}{\pi }\int _{0}^{\pi }xdx\tag {6}\\ & =\frac {1}{\pi }\frac {\pi ^{2}}{2}\nonumber \\ & =\frac {\pi }{2}\nonumber \end{align}
For \(n>0\)
\begin{align*} \int _{0}^{\pi }x\cos mxdx & =\int _{0}^{\pi }\sum _{n=1}^{\infty }b_{n}\cos mx\cos \left ( nx\right ) dx\\ & =\sum _{n=1}^{\infty }b_{n}\int _{0}^{\pi }\cos mx\cos \left ( nx\right ) dx \end{align*}
All terms in the sum vanish except for \(n=m\) . The above becomes
\begin{align} \int _{0}^{\pi }x\cos mxdx & =b_{n}\frac {\pi }{2}\nonumber \\ b_{n} & =\frac {2}{\pi }\int _{0}^{\pi }x\cos mxdx\tag {7}\end{align}
Using integration by part, \(\int _{0}^{\pi }x\cos nxdx\) , let \(u=x,dv=\cos nx,\rightarrow du=1,v=\frac {\sin nx}{n}\) , therefore
\begin{align*} \int _{0}^{\pi }x\cos nxdx & =\left ( x\frac {\sin nx}{n}\right ) _{0}^{\pi }-\frac {1}{n}\int _{0}^{\pi }\sin nxdx\\ & =\frac {1}{n}\left ( \pi \sin n\pi -0\right ) +\frac {1}{n}\left ( \frac {\cos nx}{n}\right ) _{0}^{\pi }\\ & =0+\frac {1}{n^{2}}\left ( \cos nx\right ) _{0}^{\pi }\\ & =\frac {1}{n^{2}}\left ( \cos n\pi -1\right ) \end{align*}
Hence
\begin{align*} b_{n} & =\frac {2}{\pi }\int _{0}^{\pi }x\cos mxdx\\ & =\frac {2}{\pi }\frac {\left ( -1^{n}-1\right ) }{n^{2}}\end{align*}
Therefore the final solution is, from (5), by combing all above results, becomes
\begin{align*} u\left ( x,t\right ) & =c_{0}e^{-\beta t}+\sum _{n=1}^{\infty }b_{n}\cos \left ( nx\right ) e^{-\left ( n^{2}k+\beta \right ) t}\\ & =\frac {\pi }{2}e^{-\beta t}+\frac {2}{\pi }\sum _{n=1}^{\infty }\frac {\left ( -1^{n}-1\right ) }{n^{2}}\cos \left ( nx\right ) e^{-\left ( n^{2}k+\beta \right ) t}\end{align*}
This is animation of the solution for \(k=1,\beta =2\) for \(2\) seconds. (Animation will only show in the HTML version)
Source code used for the above
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3.1.1.43 problem number 192
Added July 2, 2018.
4th example from Maple document for new improvements in Maple 2018.1, originally taken from Pinchover and Rubinstein’s exercise 6.23 .
Solve the heat equation on bar
\[ u_t= u_{xx} + g(x,t) \]
Where \(g(x,t) = e^{3 t} \cos (17 \pi x)\) for \(0<x<1\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &=0 \\ \frac {\partial u}{\partial x}(1,t) &=0 \end{align*}
Initial condition is \(u(x,0) =f(x)\) where \(f(x) = 3 \cos (42 \pi x)\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; f [ x ] := 3* Cos [42* Pi *x];
g[x, t] := Exp [3*t]* Cos [17*x* Pi ];
pde = D [u[x, t], t] == D [u[x, t], {x, 2}] + g[x, t];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][1, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\sqrt {2} \cos (\pi x K[1]) \left (\int _0^t e^{-\pi ^2 K[1]^2 (t-K[2])} \left (\begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{3 K[2]}}{\sqrt {2}} & K[1]=17 \\ 0 & \text {True} \\\end {array} \\\end {array}\right ) \, dK[2]+e^{-\pi ^2 t K[1]^2} \left (\begin {array}{cc} \{ & \begin {array}{cc} \frac {3}{\sqrt {2}} & K[1]=42 \\ 0 & \text {True} \\\end {array} \\\end {array}\right )\right )\right \}\right \}\]
Maple ✓
restart ; f := x->3* cos (42* x * Pi ); g :=(x,t)-> exp (3* t )* cos (17* x * Pi ); pde := diff ( u ( x , t), t)= ( diff ( u ( x , t), x, x)) + g(x, t);
bc := ( D [1]( u ))(0, t) = 0, ( D [1]( u ))(1, t) = 0;
ic := u(x, 0) = f(x);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , simplify ( pdsolve ([ pde , ic, bc],u(x,t)))),output= ' realtime ' ));
\[u \left (x , t\right ) = 3 \cos \left (42 \pi x \right ) {\mathrm e}^{-1764 \pi ^{2} t}+\frac {\cos \left (17 \pi x \right ) \left ({\mathrm e}^{3 t}-{\mathrm e}^{-289 \pi ^{2} t}\right )}{289 \pi ^{2}+3}\]
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3.1.1.44 problem number 193
added July 2, 2018.
Taken from Maple document for new improvements in Maple 2018.1, originally taken from Pinchover and Rubinstein’s exercise 6.21
Solve the heat equation on bar
\[ u_t = u_{xx} + g(x,t) \]
Where \(g(x,t) = t \cos (2001 x)\) for \(0<x<\pi \) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &=0 \\ \frac {\partial u}{\partial x}(1,t) &=0 \end{align*}
Initial condition is \(u(x,0) =f(x)\) where \(f(x) = \pi \cos (2 x)\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == D [u[x, t], {x, 2}] + t* Cos [2001*x];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][ Pi , t] == 0};
ic = u[x, 0] == Pi * Cos [2*x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\sqrt {\frac {2}{\pi }} \cos (x K[1]) \left (\int _0^t e^{-K[1]^2 (t-K[2])} \left (\begin {array}{cc} \{ & \begin {array}{cc} \sqrt {\frac {\pi }{2}} K[2] & K[1]=2001 \\ 0 & \text {True} \\\end {array} \\\end {array}\right ) \, dK[2]+e^{-t K[1]^2} \left (\begin {array}{cc} \{ & \begin {array}{cc} \frac {\pi ^{3/2}}{\sqrt {2}} & K[1]=2 \\ 0 & \text {True} \\\end {array} \\\end {array}\right )\right )\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t), t)= ( diff ( u ( x , t), x, x)) + t* cos (2001* x ); bc := ( D [1]( u ))(0, t) = 0, ( D [1]( u ))( Pi , t) = 0;
ic := u(x, 0) = Pi * cos (2* x ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde, ic, bc],u(x,t))),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {\left (4004001 t +{\mathrm e}^{-4004001 t}-1\right ) \cos \left (2001 x \right )}{16032024008001}+\cos \left (2 x \right ) {\mathrm e}^{-4 t} \pi \]
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3.1.1.45 problem number 194
Added April 20, 2019.
Solve the parabolic pde for \(u(x,t)\)
\[ u_t = x^2 u_{xx} + x u_x \]
For \(1<x<b\) and \(t>0\) . The boundary conditions are
\begin{align*} u_x(1,t) &= 0\\ h u(b,t) + u_x(b,t) &= 0 \end{align*}
Where \(h>0\) . Initial condition is \(u(x,0)=\ln x\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == x^2* D [u[x, t], {x, 2}] + x* D [u[x, t], x];
ic = u[x, 0] == Log [x];
bc = { Derivative [1, 0][u][1, t] == 0, h*u[b, t] + Derivative [1, 0][u][b, t] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], {x, t}, Assumptions -> {b > 1, t > 0, h > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {2 b e^{-t K[2,K[1]]} h \cos \left (\sqrt {K[2,K[1]]} \log (x)\right ) \left (\cos \left (\sqrt {K[2,K[1]]} \log (b)\right )+\sqrt {K[2,K[1]]} \log (b) \sin \left (\sqrt {K[2,K[1]]} \log (b)\right )-1\right )}{K[2,K[1]] \left (\sin ^2\left (\sqrt {K[2,K[1]]} \log (b)\right )+b h \log (b)\right )} & \frac {b h}{\sqrt {K[2,K[1]]}}=\tan \left (\sqrt {K[2,K[1]]} \log (b)\right )\land K[1]\in \mathbb {Z}\land K[1]\geq 1\land K[2,K[1]]>0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✗
restart ; pde := diff ( u ( x , t ), t )= x ^2* diff ( u ( x , t ), x$2 )+ x * diff ( u ( x , t ), x ); bc := eval ( diff ( u ( x , t ), x ), x =1)=0, h * u ( b , t )+ eval ( diff ( u ( x , t ), x ), x = b )=0; ic := u(x,0)= ln ( x ); cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve({pde, ic, bc}, u(x, t))),output= ' realtime ' ));
time expired
Hand
solution
Solve
\begin{equation} u_{t}=x^{2}u_{xx}+xu_{x}\tag {1}\end{equation}
\(1<x\,<b,t>0\) . BC
\begin{align*} u_{x}\left ( 1,t\right ) & =0\\ hu\left ( b,t\right ) +u_{x}\left ( b,t\right ) & =0 \end{align*}
Where \(h>0\) . And initial conditions
\[ u\left ( x,0\right ) =\ln x \]
Let \(u=X\left ( x\right ) T\left ( x\right ) \) . Substituting into (1) gives
\begin{align*} T^{\prime }X & =x^{2}X^{\prime \prime }T+xX^{\prime }T\\ \frac {T^{\prime }}{T} & =x^{2}\frac {X^{\prime \prime }}{X}+x\frac {X^{\prime }}{X}=-\lambda \end{align*}
Where \(\lambda \) is the separation constant. From the boundary conditions, we know that \(\lambda \) will be only positive. So we do not need to check for possibility of negative of zero eigenvalue. Letting \(\lambda =\alpha ^{2}\) , then the above reduces to
\begin{align} x^{2}X^{\prime \prime }+xX^{\prime }+\alpha ^{2}X & =0\tag {2}\\ T^{\prime }+\alpha ^{2}T & =0\tag {3}\end{align}
Equation (2) is Euler ODE. Assuming \(X=x^{r}\) , then \(X^{\prime }=rx^{r-1},X^{\prime \prime }=r\left ( r-1\right ) x^{r-2}\) . Substituting back into (2) gives the characteristic equation
\begin{align*} r\left ( r-1\right ) x^{r}+rx^{r}+\alpha ^{2}x^{r} & =0\\ r^{2}+\alpha ^{2} & =0\\ r & =\pm i\alpha \end{align*}
Hence the solution to (2) is
\begin{align*} X\left ( x\right ) & =Ax^{i\alpha }+Bx^{-i\alpha }\\ & =Ae^{\ln x^{i\alpha }}+Be^{\ln x^{-i\alpha }}\\ & =Ae^{i\alpha \ln x}+Be^{-i\alpha \ln x}\end{align*}
Which using Euler relation can be written as (using new constants, but the name of the constants kept the same for simplicity)
\[ X\left ( x\right ) =A\cos \left ( \alpha \ln x\right ) +B\sin \left ( \alpha \ln x\right ) \]
Applying first BC. \(X^{\prime }\left ( 1,t\right ) =0\) gives
\begin{align*} X^{\prime }\left ( x\right ) & =-\frac {\alpha }{x}A\sin \left ( \alpha \ln x\right ) +B\frac {\alpha }{x}\cos \left ( \alpha \ln x\right ) \\ 0 & =-\alpha A\sin \left ( \alpha \ln 1\right ) +B\alpha \cos \left ( \alpha \ln 1\right ) \\ & =B\alpha \end{align*}
Since \(\alpha >0\) then \(B=0\) . Hence the solution becomes
\[ X\left ( x\right ) =A\cos \left ( \alpha \ln x\right ) \]
Applying second BC. \(hX\left ( b\right ) +X^{\prime }\left ( b\right ) =0\) gives
\begin{align} hA\cos \left ( \alpha \ln b\right ) -A\frac {\alpha }{b}\sin \left ( \alpha \ln b\right ) & =0\nonumber \\ h-\frac {\alpha }{b}\tan \left ( \alpha \ln b\right ) & =0\nonumber \\ \tan \left ( \alpha \ln b\right ) & =\frac {hb}{\alpha }\tag {4}\end{align}
There is no analytical solution to the above. The eigenvalues \(\alpha _{n}\) are the solutions to the above nonlinear equation. Therefore the eigenfunctions are
\[ X_{n}\left ( x\right ) =\cos \left ( \alpha _{n}\ln x\right ) \]
With eigenvalues \(\alpha _{n}>0\) given by solutions to (4). The solution to the time ODE is
\begin{align*} T_{n}^{\prime }+\alpha _{n}^{2}T_{n} & =0\\ T_{n}\left ( t\right ) & =T_{n}\left ( 0\right ) e^{-\alpha _{n}^{2}t}\end{align*}
Hence the solution to (1) is
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }T_{n}\left ( 0\right ) e^{-\alpha _{n}^{2}t}\cos \left ( \alpha _{n}\ln x\right ) \tag {5}\end{equation}
\(t=0\)
\[ \ln x=\sum _{n=1}^{\infty }T_{n}\left ( 0\right ) \cos \left ( \alpha _{n}\ln x\right ) \]
Applying orthogonality gives
\begin{align*} \int _{1}^{b}\ln x\cos \left ( \alpha _{n}\ln x\right ) dx & =T_{n}\left ( 0\right ) \int _{1}^{b}\cos ^{2}\left ( \alpha _{n}\ln x\right ) dx\\ T_{n}\left ( 0\right ) & =\frac {\int _{1}^{b}\ln x\cos \left ( \alpha _{n}\ln x\right ) dx}{\int _{1}^{b}\cos ^{2}\left ( \alpha _{n}\ln x\right ) dx}\end{align*}
Hence the solution (5) becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }\left ( \frac {\int _{1}^{b}\ln x\cos \left ( \alpha _{n}\ln x\right ) dx}{\int _{1}^{b}\cos ^{2}\left ( \alpha _{n}\ln x\right ) dx}\right ) e^{-\alpha _{n}^{2}t}\cos \left ( \alpha _{n}\ln x\right ) \]
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3.1.1.46 problem number 195
Added April 28, 2019
Problem 4, section 74, Fourier series and Boundary value problem, 8th edition by Brown and Churchill.
Solve the heat equation
\[ u_t = k u_{xx} \]
For \(0<x<1,t>0\) . The boundary conditions are, on the left end \(u(0,t)=0\) and on the right end \(u_x(1,t)=-h u(1,t)\) with \(h>0\) . Initial conditions \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
ic = u[x, 0] == f[x];
bc = {u[0, t] == 0, Derivative [1, 0][u][1, t] == -h u[1, t]};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {k > 0, t > 0, h > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \underset {K[1]=1}{\overset {\infty }{\sum }}-\frac {\sqrt {2} e^{-k t K[2,K[1]]} \sqrt {h} \left (\int _0^1 -\frac {2 \sqrt {h} f(x) \sin \left (x \sqrt {K[2,K[1]]}\right )}{\sqrt {2 h+\cos \left (2 \sqrt {K[2,K[1]]}\right )+1}} \, dx\right ) \sin \left (x \sqrt {K[2,K[1]]}\right )}{\sqrt {\cos ^2\left (\sqrt {K[2,K[1]]}\right )+h}} & h \tan \left (\sqrt {K[2,K[1]]}\right )+\sqrt {K[2,K[1]]}=0\land K[1]\in \mathbb {Z}\land K[1]\geq 1\land K[2,K[1]]>0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , t ), t) = k*( diff ( u ( x , t ), x, x));
ic := u(x,0) = f(x);
bc := u(0,t)=0, eval ( diff ( u ( x , t ), x), x = 1)=-h*u(1,t);
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc , ic ], u ( x , t )) assuming k>0,h>0,t>0),output= ' realtime ' ));
\[u \left (x , t\right ) = -2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}\frac {h \sin \left (h \tan \left (\lambda _{n}\right ) x \right ) {\mathrm e}^{-k \tan \left (\lambda _{n}\right )^{2} h^{2} t} \int _{0}^{1}f \left (x \right ) \sin \left (h \tan \left (\lambda _{n}\right ) x \right )d x {| \tan \left (\lambda _{n}\right )|}}{-h {| \tan \left (\lambda _{n}\right )|}+\cos \left (h \tan \left (\lambda _{n}\right )\right ) \sin \left (h {| \tan \left (\lambda _{n}\right )|}\right )}\right )\boldsymbol {\operatorname {where}}\left \{\lambda _{n}-h \,\operatorname {csgn}\left (\tan \left (\lambda _{n}\right )\right ) \tan \left (\lambda _{n}\right )=0\wedge -\infty \le \lambda _{n}\le \infty \right \}\]
Hand
solution
To solve the PDE, we first check the boundary conditions by writing them as
\begin{align*} a_{1}u\left ( 0,t\right ) +a_{2}u_{x}\left ( 0,t\right ) & =0\\ b_{1}u\left ( 1,t\right ) +b_{2}u_{x}\left ( 1,t\right ) & =0 \end{align*}
Then \(a_{1}=0,a_{2}=0\) . Hence \(a_{1}a_{2}=0\) . And \(b_{1}=1,b_{2}=h\) . Then since it is assumed that \(h>0\) , then \(b_{1}b_{2}\geq 0\) . And since \(q\left ( x\right ) =0\) from the PDE itself, then we know that eigenvalues must be \(\lambda \geq 0\) .
Let \(u=X\left ( x\right ) T\left ( t\right ) \) then the PDE becomes
\begin{align*} T^{\prime }X & =X^{\prime \prime }T\\ \frac {T^{\prime }}{T} & =\frac {X^{\prime \prime }}{X}=-\lambda \end{align*}
Hence the Sturm Liouville problem is
\begin{align*} X^{\prime \prime }+\lambda X & =0\\ X\left ( 0\right ) & =0\\ X^{\prime }\left ( 1\right ) +hX\left ( 1\right ) & =0 \end{align*}
Where \(p\left ( x\right ) =1\) .
Case \(\lambda =0\)
Solution is
\[ X\left ( x\right ) =Ax+B \]
At \(x=0\)
\[ 0=B \]
Hence solution becomes
\[ X\left ( x\right ) =Ax \]
At \(x=1\) the second boundary conditions gives
\begin{align*} A+hA & =0\\ A\left ( 1+h\right ) & =0 \end{align*}
For non trivial solution \(1+h=0\) or \(h=-1\) . But we assumed that \(h>0\) . Therefore \(\lambda =0\) is not eigenvalue.
Case \(\lambda >0\)
Let \(\lambda =\alpha ^{2},\alpha >0\) . Hence solution is
\[ X\left ( x\right ) =A\cos \left ( \alpha x\right ) +B\sin \left ( \alpha x\right ) \]
At \(X\left ( 0\right ) =0\)
\[ 0=A \]
The solution becomes
\[ X\left ( x\right ) =B\sin \left ( \alpha x\right ) \]
At \(x=1\) the second boundary conditions gives
\begin{align*} B\alpha \cos \left ( \alpha \right ) +hB\sin \left ( \alpha \right ) & =0\\ \alpha \cos \left ( \alpha \right ) +h\sin \left ( \alpha \right ) & =0\\ \tan \left ( \alpha \right ) & =-\frac {\alpha }{h}\end{align*}
Therefore the eigenvalues are given by solution to
\[ \tan \left ( \alpha _{n}\right ) =-\frac {\alpha _{n}}{h}\qquad n=1,2,3,\cdots \]
And eigenfunctions are
\[ X_{n}\left ( x\right ) =\sin \left ( \alpha _{n}x\right ) \]
The normalized eigenfunctions are
\[ \phi _{n}\left ( x\right ) =\frac {X_{n}\left ( x\right ) }{\left \Vert X_{n}\left ( x\right ) \right \Vert }\]
But
\begin{align*} \left \Vert X_{n}\left ( x\right ) \right \Vert ^{2} & =\int _{0}^{1}p\left ( x\right ) X_{n}^{2}\left ( x\right ) dx\\ & =\int _{0}^{1}\sin ^{2}\left ( \alpha _{n}x\right ) dx\\ & =\frac {1}{2}\int _{0}^{1}1-\cos \left ( 2\alpha _{n}x\right ) dx\\ & =\frac {1}{2}\left ( 1-\left [ \frac {\sin \left ( 2\alpha _{n}x\right ) }{2\alpha _{n}}\right ] _{0}^{1}\right ) \\ & =\frac {1}{2}\left ( 1-\frac {1}{2\alpha _{n}}\left [ \sin \left ( 2\alpha _{n}x\right ) \right ] _{0}^{1}\right ) \\ & =\frac {1}{2}\left ( 1-\frac {\sin \left ( 2\alpha _{n}\right ) }{2\alpha _{n}}\right ) \\ & =\frac {1}{2}-\frac {\sin \left ( 2\alpha _{n}\right ) }{4\alpha _{n}}\end{align*}
But \(\sin \left ( 2\alpha _{n}\right ) =2\sin \alpha _{n}\cos \alpha _{n}\) and \(\alpha _{n}=-h\frac {\sin \left ( \alpha _{n}\right ) }{\cos \left ( \alpha _{n}\right ) }\) , therefore the above becomes
\begin{align*} \left \Vert X_{n}\left ( x\right ) \right \Vert ^{2} & =\frac {1}{2}+\frac {2\sin \alpha _{n}\cos \alpha _{n}}{4h\frac {\sin \left ( \alpha _{n}\right ) }{\cos \left ( \alpha _{n}\right ) }}\\ & =\frac {1}{2}+\frac {\cos ^{2}\alpha _{n}}{2h}\\ & =\frac {h+\cos ^{2}\alpha _{n}}{2h}\end{align*}
Hence
\begin{align*} \phi _{n}\left ( x\right ) & =\frac {X_{n}\left ( x\right ) }{\sqrt {\frac {h+\cos ^{2}\alpha _{n}}{2h}}}\\ & =\sqrt {\frac {2h}{h+\cos ^{2}\alpha _{n}}}\sin \left ( \alpha _{n}x\right ) \end{align*}
Now we use generalized Fourier series to find the solution. Let
\begin{equation} u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\left ( t\right ) \phi _{n}\left ( x\right ) \tag {1}\end{equation}
\[ \sum _{n=1}^{\infty }B_{n}^{\prime }\left ( t\right ) \phi _{n}\left ( x\right ) =k\sum _{n=1}^{\infty }B_{n}\left ( t\right ) \phi _{n}^{\prime \prime }\left ( x\right ) \]
But \(\phi _{n}^{\prime \prime }\left ( x\right ) =-\lambda _{n}\phi _{n}\left ( x\right ) =-\alpha _{n}^{2}\phi _{n}\left ( x\right ) \) . The above becomes
\begin{align*} \sum _{n=1}^{\infty }B_{n}^{\prime }\left ( t\right ) \phi _{n}\left ( x\right ) & =-k\sum _{n=1}^{\infty }B_{n}\left ( t\right ) \alpha _{n}^{2}\phi _{n}\left ( x\right ) \\ B_{n}^{\prime }\left ( t\right ) +k\alpha _{n}^{2}B_{n}\left ( t\right ) & =0 \end{align*}
The solution is
\[ B_{n}\left ( t\right ) =B_{n}\left ( 0\right ) e^{-k\alpha _{n}^{2}t}\]
Hence (1) becomes
\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\left ( 0\right ) e^{-k\alpha _{n}^{2}t}\phi _{n}\left ( x\right ) \]
At \(t=0\) the above becomes
\[ f\left ( x\right ) =\sum _{n=1}^{\infty }B_{n}\left ( 0\right ) \phi _{n}\left ( x\right ) \]
Therefore
\begin{align*} B_{n}\left ( 0\right ) & =\left \langle f\left ( x\right ) ,\phi _{n}\left ( x\right ) \right \rangle \\ & =\int _{0}^{1}p\left ( x\right ) f\left ( x\right ) \phi _{n}\left ( x\right ) dx\\ & =\sqrt {\frac {2h}{h+\cos ^{2}\alpha _{n}}}\int _{0}^{1}f\left ( x\right ) \sin \left ( \alpha _{n}x\right ) dx \end{align*}
Therefore
\begin{align*} B_{n}\left ( t\right ) & =B_{n}\left ( 0\right ) e^{-k\alpha _{n}^{2}t}\\ & =\left ( \sqrt {\frac {2h}{h+\cos ^{2}\alpha _{n}}}\int _{0}^{1}f\left ( x\right ) \sin \left ( \alpha _{n}x\right ) dx\right ) e^{-k\alpha _{n}^{2}t}\end{align*}
and solution (1) becomes
\begin{align*} u\left ( x,t\right ) & =\sum _{n=1}^{\infty }\sqrt {\frac {2h}{h+\cos ^{2}\alpha _{n}}}\left ( \int _{0}^{1}f\left ( x\right ) \sin \left ( \alpha _{n}x\right ) dx\right ) e^{-k\alpha _{n}^{2}t}\sqrt {\frac {2h}{h+\cos ^{2}\alpha _{n}}}\sin \left ( \alpha _{n}x\right ) \\ & =\frac {2h}{h+\cos ^{2}\alpha _{n}}\sum _{n=1}^{\infty }\left ( \int _{0}^{1}f\left ( x\right ) \sin \left ( \alpha _{n}x\right ) dx\right ) e^{-k\alpha _{n}^{2}t}\sin \left ( \alpha _{n}x\right ) \end{align*}
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3.1.1.47 problem number 196
Solve the heat equation
\[ \frac { \partial u}{\partial t} = k \frac { \partial ^2 u}{\partial x^2} \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac { \partial u}{\partial x}(0,t) +u(0,t) &=0 \\ \frac { \partial u}{\partial x}(L,t) +u(L,t) &=0 \end{align*}
And initial condition \(u(x,0)=f(x)\)
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}];
bc = { Derivative [1, 0][u][0, t] + u[0, t] == 0, Derivative [1, 0][u][L, t] + u[L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc, ic}, u[x, t], {x, t}, Assumptions -> {k > 0, x > 0, x < L}], 60*10]];
\[\left \{\left \{u(x,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} e^{-k t K[2,K[1]]} \left (\int _0^L \frac {\sqrt {2} f(x) \sqrt {K[2,K[1]]+1} \left (\cos \left (x \sqrt {K[2,K[1]]}\right ) \sqrt {K[2,K[1]]}-\sin \left (x \sqrt {K[2,K[1]]}\right )\right )}{\sqrt {(K[2,K[1]]+1) \left (K[2,K[1]] L+L+\cos \left (2 L \sqrt {K[2,K[1]]}\right )-1\right )}} \, dx\right ) \sqrt {K[2,K[1]]+1} \left (\cos \left (x \sqrt {K[2,K[1]]}\right ) \sqrt {K[2,K[1]]}-\sin \left (x \sqrt {K[2,K[1]]}\right )\right )}{\sqrt {2 K[2,K[1]] \cos ^2\left (L \sqrt {K[2,K[1]]}\right )+2 \cos ^2\left (L \sqrt {K[2,K[1]]}\right )+L K[2,K[1]]^2+L+2 L K[2,K[1]]-2 K[2,K[1]]-2}} & (K[2,K[1]]+1) \sin \left (L \sqrt {K[2,K[1]]}\right )=0\land K[1]\in \mathbb {Z}\land K[1]\geq 1\land K[2,K[1]]>0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 ); ic := u(x,0)=f(x);
bc := D [1]( u )(0, t )+ u (0, t )=0, D [1]( u )( L , t )+ u ( L , t )=0; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , ic , bc ], u ( x , t )) assuming k>0,x>0,x<L),output= ' realtime ' ));
\[u \left (x , t\right ) = 2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {\left (-\pi n \cos \left (\frac {n \pi x}{L}\right )+\sin \left (\frac {n \pi x}{L}\right ) L \right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}f \left (x \right ) \left (-\pi n \cos \left (\frac {n \pi x}{L}\right )+\sin \left (\frac {n \pi x}{L}\right ) L \right )d x}{L \left (\pi ^{2} n^{2}+L^{2}\right )}\right )\]
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3.1.1.48 problem number 197
added March 18, 2018.
This is problem 8.2.1, part(f) from Richard Haberman applied partial differential equations 5th edition.
Solve the heat equation
\[ \frac { \partial u}{\partial t}= k \frac { \partial ^2 u}{\partial x^2} + \sin \left ( \frac {2 \pi x}{L} \right ) \]
For \(0<x<L\) and \(t>0\) . The boundary conditions are
\begin{align*} \frac {\partial u}{\partial x}(0,t) &= 0 \\ \frac {\partial u}{\partial x}(L,t) &= 0 \\ \end{align*}
Initial condition is \(u(x,0)=f(x)\) .
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = D [u[x, t], t] == k* D [u[x, t], {x, 2}] + Sin [(2* Pi *x)/L];
bc = { Derivative [1, 0][u][0, t] == 0, Derivative [1, 0][u][L, t] == 0};
ic = u[x, 0] == f[x];
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, ic, bc}, u[x, t], x, t, Assumptions -> {L > 0, k > 0, t > 0}], 60*10]];
\[\left \{\left \{u(x,t)\to \underset {K[1]=1}{\overset {\infty }{\sum }}\frac {\sqrt {2} \cos \left (\frac {\pi x K[1]}{L}\right ) \left (\frac {\left (1-e^{-\frac {k \pi ^2 t K[1]^2}{L^2}}\right ) \left (\begin {array}{cc} \{ & \begin {array}{cc} 0 & K[1]=2 \\ \frac {2 \sqrt {2} \left (-1+(-1)^{K[1]}\right ) \sqrt {L}}{\pi \left (K[1]^2-4\right )} & \text {True} \\\end {array} \\\end {array}\right ) L^2}{k \pi ^2 K[1]^2}+e^{-\frac {k \pi ^2 t K[1]^2}{L^2}} \int _0^L \frac {\sqrt {2} \cos \left (\frac {\pi x K[1]}{L}\right ) f(x)}{\sqrt {L}} \, dx\right )}{\sqrt {L}}+\frac {\int _0^L \frac {f(x)}{\sqrt {L}} \, dx}{\sqrt {L}}\right \}\right \}\]
Maple ✓
restart ; interface ( showassumed =0); pde := diff ( u ( x , t ), t )= k * diff ( u ( x , t ), x$2 )+ sin (2* Pi * x / L ); ic := u(x,0)=f(x);
bc := D [1]( u )(0, t )=0, D [1]( u )( L , t )=0; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve([pde,ic,bc],u(x,t)) assuming L>0,t>0,k>0),output= ' realtime ' ));
\[u \left (x , t\right ) = \frac {4 c_{2} \pi ^{2} k L +L^{3} \sin \left (\frac {2 \pi x}{L}\right )-2 L^{2} x \pi -2 \left (\moverset {\infty }{\munderset {n =1}{\sum }}\cos \left (\frac {n \pi x}{L}\right ) {\mathrm e}^{-\frac {k \,\pi ^{2} n^{2} t}{L^{2}}} \int _{0}^{L}\left (-4 f \left (x \right ) \pi ^{2} k +4 c_{2} \pi ^{2} k +L^{2} \sin \left (\frac {2 \pi x}{L}\right )-2 L x \pi \right ) \cos \left (\frac {n \pi x}{L}\right )d x \right )-\int _{0}^{L}\left (-4 f \left (x \right ) \pi ^{2} k +4 c_{2} \pi ^{2} k +L^{2} \sin \left (\frac {2 \pi x}{L}\right )-2 L x \pi \right )d x}{4 \pi ^{2} k L}\]
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