4.3.1
4.3.1.1 problem number 326
Added March 13, 2019.
Solve for \(u(x,y) \)
\begin{align*} \frac {u_{xx}}{A} + \frac {u_{xx}}{B} = -2 \theta \end{align*}
Where \(A,B,\theta \) are constants, and the boundary conditions are
\begin{align*} u(x, -b) &= 0\\ u(x, b) &= 0\\ u(-a, y) &= 0\\ u(a, y) &= 0 \end{align*}
Mathematica ✗
ClearAll [ " Global ` * " ]; pde = D [u[x, y], {x, 2}]/A + D [u[x, y], {y, 2}]/B == -2*theta;
bc = {u[x, -b] == 0, u[x, b] == 0, u[-a, y] == 0, u[a, y] == 0};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc}, u[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart ; pde := diff ( u ( x , y ), x$2 )/ A + diff ( u ( x , y ), y$2 )/ B = -2*theta;
bc := u(x,-b)=0, u(x,b)=0, u(-a,y)=0, u(a,y) = 0;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc ], u ( x , y ))), output = ' realtime ' )); \[u \left (x , y\right ) = \left (\moverset {\infty }{\munderset {n =1}{\sum }}\left \{\begin {array}{cc} \operatorname {csgn}\left (\frac {1}{a}\right ) \sin \left (\frac {n \pi x}{a}\right ) \left ({\mathrm e}^{-\frac {n \pi \,\operatorname {csgn}\left (\frac {1}{a}\right ) \sqrt {B}\, y}{a \sqrt {A}}} f_{2} \left (n \right )+{\mathrm e}^{\frac {n \pi \,\operatorname {csgn}\left (\frac {1}{a}\right ) \sqrt {B}\, y}{a \sqrt {A}}} f_{1} \left (n \right )\right ) & a =0 \\ 0 & \operatorname {otherwise} \end {array}\right .\right )+\theta A \left (a^{2}-x^{2}\right )\]
Hand
solution
solve
\begin{align*} \frac {u_{xx}}{A}+\frac {u_{yy}}{B} & =-2\theta \\ Bu_{xx}+Au_{yy} & =-2\theta AB\\ & =C \end{align*}
Where \(C=-2\theta AB\) is a new constant. With boundary conditions
\begin{align*} u\left ( x,-b\right ) & =0\\ u\left ( x,b\right ) & =0\\ u\left ( -a,y\right ) & =0\\ u\left ( a,y\right ) & =0 \end{align*}
To simplify solution, shift the rectangle so its lower left corner on the origin. Let \(\tilde {x}=x+a\) , and \(\tilde {y}=y+b\) . The boundary conditions becomes
\begin{align*} u\left ( \tilde {x},0\right ) & =0\\ u\left ( \tilde {x},2b\right ) & =0\\ u\left ( 0,\tilde {y}\right ) & =0\\ u\left ( 2a,\tilde {y}\right ) & =0 \end{align*}
And the pde becomes \(Bu_{\tilde {x}\tilde {x}}+Au_{\tilde {y}\tilde {y}}=C\) . Instead of keep writing \(\tilde {x},\tilde {y}\) , will use \(x,y\) , but remember that these are shifted version. At the end, we shift back.
Hence the PDE to solve is \(Bu_{xx}+Au_{yy}=C\) with BC
\begin{align*} u\left ( x,0\right ) & =0\\ u\left ( x,2b\right ) & =0\\ u\left ( 0,y\right ) & =0\\ u\left ( 2a,y\right ) & =0 \end{align*}
Using eigenfunction expansion method. Let
\begin{equation} u\left ( x,y\right ) =\sum _{n=1}^{\infty }b_{n}\left ( y\right ) X_{n}\left ( x\right ) \tag {1}\end{equation}
\(X_{n}\left ( x\right ) \) is eigenfunctions for \(X^{\prime \prime }\left ( x\right ) +\lambda _{n}X\left ( x\right ) =0\) with boundary conditions \(X\left ( 0\right ) =X\left ( 2a\right ) =0\) . This has eigenfunctions as \(X_{n}\left ( x\right ) =\sin \left ( \sqrt {\lambda _{n}}x\right ) \) with eigenvalues \(\lambda _{n}=\left ( \frac {n\pi }{2a}\right ) ^{2}\) for \(n=1,2,\cdots \) .
Substituting (1) into the PDE \(Bu_{xx}+Au_{yy}=C\) gives
\[ B\sum _{n=1}^{\infty }b_{n}\left ( y\right ) X_{n}^{\prime \prime }\left ( x\right ) +A\sum _{n=1}^{\infty }b_{n}^{\prime \prime }\left ( y\right ) X_{n}\left ( x\right ) =C \]
Expanding \(C\) (a constant) as Fourier sine series the above becomes
\[ B\sum _{n=1}^{\infty }b_{n}\left ( y\right ) X_{n}^{\prime \prime }\left ( x\right ) +A\sum _{n=1}^{\infty }b_{n}^{\prime \prime }\left ( y\right ) X_{n}\left ( x\right ) =\sum _{n=1}^{\infty }q_{n}X_{n}\left ( x\right ) \]
But \(X_{n}^{\prime \prime }\left ( x\right ) =-\lambda _{n}X_{n}\left ( x\right ) \) , hence the above becomes
\begin{align} -B\sum _{n=1}^{\infty }\lambda _{n}b_{n}\left ( y\right ) X_{n}\left ( x\right ) +A\sum _{n=1}^{\infty }b_{n}^{\prime \prime }\left ( y\right ) X_{n}\left ( x\right ) & =\sum _{n=1}^{\infty }q_{n}X_{n}\left ( x\right ) \nonumber \\ Ab_{n}^{\prime \prime }\left ( y\right ) -B\lambda _{n}b_{n}\left ( y\right ) & =q_{n}\tag {1A}\end{align}
But
\begin{align*} C & =\sum _{n=1}^{\infty }q_{n}X_{n}\left ( x\right ) \\ \int _{0}^{2a}CX_{n}\left ( x\right ) dx & =q_{n}\int _{0}^{2a}X_{n}^{2}\left ( x\right ) dx\\ \int _{0}^{2a}C\sin \left ( \sqrt {\lambda _{n}}x\right ) dx & =q_{n}\int _{0}^{2a}\sin ^{2}\left ( \sqrt {\lambda _{n}}x\right ) dx\\ \frac {-C}{\sqrt {\lambda _{n}}}\left ( \left ( -1\right ) ^{n}-1\right ) & =q_{n}a\\ q_{n} & =\frac {-C}{a\sqrt {\lambda _{n}}}\left ( \left ( -1\right ) ^{n}-1\right ) \end{align*}
Hence (1A) becomes
\[ Ab_{n}^{\prime \prime }\left ( y\right ) -B\lambda _{n}b_{n}\left ( y\right ) =\frac {-C}{a\sqrt {\lambda _{n}}}\left ( \left ( -1\right ) ^{n}-1\right ) \]
This is standard second order linear ODE. The solution is
\[ b_{n}\left ( y\right ) =D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}+\frac {C}{aB\lambda _{n}^{\frac {3}{2}}}\left ( \left ( -1\right ) ^{n}-1\right ) \]
Using the above in (1) gives the solution
\begin{equation} u\left ( x,y\right ) =\sum _{n=1}^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}+\frac {C}{aB\lambda _{n}^{\frac {3}{2}}}\left ( \left ( -1\right ) ^{n}-1\right ) \right ) X_{n}\left ( x\right ) \tag {1A}\end{equation}
\(D_{n},E_{n}\) .
Case \(n\) even
When \(n\) is even \(\left ( \left ( -1\right ) ^{n}-1\right ) =0\) and the solution (1A) becomes
\[ u\left ( x,y\right ) =\sum _{n=1}^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}\right ) X_{n}\left ( x\right ) \]
At \(y=0\) the above gives
\[ 0=\sum _{n=1}^{\infty }\left ( D_{n}+E_{n}\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Therefore
\begin{equation} D_{n}+E_{n}=0\tag {2}\end{equation}
\(y=2b\)
\[ 0=\sum _{n=1}^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}2b}\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Therefore
\begin{equation} D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}2b}=0\tag {3}\end{equation}
\(D_{n}=E_{n}=0\) , Hence \(u\left ( x,y\right ) =0\) when \(n\) even.
Case \(n\) odd
When \(n\) is odd \(\left ( \left ( -1\right ) ^{n}-1\right ) =-2\) and the solution (1A) becomes
\[ u\left ( x,y\right ) =\sum _{n=1}^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) X_{n}\left ( x\right ) \]
At \(y=0\) the above gives
\[ 0=\sum _{n=1}^{\infty }\left ( D_{n}+E_{n}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Therefore
\begin{equation} D_{n}+E_{n}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}=0\tag {4}\end{equation}
\(y=2b\)
\[ 0=\sum _{n=1}^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}2b}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \]
Therefore
\begin{equation} D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}2b}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}=0\tag {5}\end{equation}
\(D_{n},E_{n}\) gives
\begin{align*} D_{n} & =\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {1}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\\ E_{n} & =\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\end{align*}
Therefore the final solution from (1A) becomes
\begin{align*} u\left ( x,y\right ) & =\sum _{n=1,3,5,\cdots }^{\infty }\left ( D_{n}e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+E_{n}e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) X_{n}\left ( x\right ) \\ & =\sum _{n=1,3,5,\cdots }^{\infty }\left ( \left ( \frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {1}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{\sqrt {\frac {B}{A}\lambda _{n}}y}+\left ( \frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{-\sqrt {\frac {B}{A}\lambda _{n}}y}-\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}x\right ) \end{align*}
Where \(\lambda _{n}=\left ( \frac {n\pi }{2a}\right ) ^{2}\) . Switching back to original coordinates using \(\tilde {x}=x+a\) , and \(\tilde {y}=y+b\) , then the above is
\[ u\left ( x,y\right ) =\sum _{n=1,3,5,\cdots }^{\infty }\left ( \left ( \frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {1}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{\sqrt {\frac {B}{A}\lambda _{n}}\left ( y+b\right ) }+\left ( \frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\frac {e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e\left ( ^{-\sqrt {\frac {B}{A}\lambda _{n}}y+b}\right ) -\frac {2C}{aB\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}\left ( x+a\right ) \right ) \]
Where \(C=-2\theta AB\) , hence
\begin{align*} u\left ( x,y\right ) & =\sum _{n=1,3,5,\cdots }^{\infty }\left ( \left ( \frac {-4\theta AB}{aB\lambda _{n}^{\frac {3}{2}}}\frac {1}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{\sqrt {\frac {B}{A}\lambda _{n}}\left ( y+b\right ) }+\left ( \frac {-4\theta AB}{aB\lambda _{n}^{\frac {3}{2}}}\frac {e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{-\sqrt {\frac {B}{A}\lambda _{n}}\left ( y+b\right ) }+\frac {4\theta AB}{aB\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}\left ( x+a\right ) \right ) \\ & =\sum _{n=1,3,5,\cdots }^{\infty }\left ( \left ( \frac {-4\theta A}{a\lambda _{n}^{\frac {3}{2}}}\frac {1}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{\sqrt {\frac {B}{A}\lambda _{n}}\left ( y+b\right ) }+\left ( \frac {-4\theta A}{a\lambda _{n}^{\frac {3}{2}}}\frac {e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}{1+e^{\sqrt {\frac {B}{A}\lambda _{n}}2b}}\right ) e^{-\sqrt {\frac {B}{A}\lambda _{n}}\left ( y+b\right ) }+\frac {4\theta A}{a\lambda _{n}^{\frac {3}{2}}}\right ) \sin \left ( \sqrt {\lambda _{n}}\left ( x+a\right ) \right ) \end{align*}
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4.3.1.2 problem number 327
Taken from Mathematica DSolve help pages.
Solve for \(u\left ( x,y\right ) \)
\begin{align*} u_{xx}+ u_{yy} & = 6x - 6y \end{align*}
Boundary conditions
\begin{align*} u(x, 0) &= 1 + 11 x + x^3\\ u(x, 2) &= -7 + 11 x + x^3\\ u(0, y) &= 1 - y^3\\ u(4, y) &= 109 - y^3 \end{align*}
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = Laplacian[u[x, y], {x, y}] == 6*x - 6*y;
bc = {u[x, 0] == 1 + 11*x + x^3, u[x, 2] == -7 + 11*x + x^3, u[0, y] == 1 - y^3, u[4, y] == 109 - y^3};
sol = AbsoluteTiming [ TimeConstrained [ DSolve [{pde, bc}, u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to x^3+11 x-y^3+1\right \}\right \}\]
Maple ✓
restart ; pde := diff ( u ( x , y ), x$2 )+ diff ( u ( x , y ), y$2 )=6* x -6* y ; bc := u(x,0)=1+11*x+x^3,
u(x,2)=-7+11*x+x^3,
u(0,y)=1-y^3,
u(4,y)=109-y^3;
cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ([ pde , bc ], u ( x , y ))), output = ' realtime ' )); \[u \left (x , y\right ) = x^{3}-y^{3}+11 x +1\]
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4.3.1.3 problem number 328
Added January 13, 2020
Solve Poisson PDE
\[ \frac {\partial ^{2}f}{\partial x^{2}}+\frac {\partial ^{2}f}{\partial y^{2}}=6y \]
Mathematica ✓
ClearAll [ " Global ` * " ]; pde = Laplacian[u[x, y], {x, y}] == 6*y;
sol = AbsoluteTiming [ TimeConstrained [ DSolve [pde, u[x, y], {x, y}], 60*10]];
\[\left \{\left \{u(x,y)\to c_1(x-i y)+c_2(x+i y)+3 x^2 y\right \}\right \}\]
Maple ✓
restart ; pde := VectorCalculus :- Laplacian ( u ( x , y ), ' cartesian ' [ x , y ])=6* y ; cpu_time := timelimit (60*10, CodeTools [ Usage ]( assign ( ' sol ' , pdsolve ( pde , u ( x , y ))), output = ' realtime ' )); \[u \left (x , y\right ) = f_{1} \left (-i x +y \right )+f_{2} \left (i x +y \right )+3 x^{2} y\]
Hand
solution
Solve Poisson PDE
\[ \frac {\partial ^{2}f}{\partial x^{2}}+\frac {\partial ^{2}f}{\partial y^{2}}=6y \]
The solution is
\begin{equation} f=f_{h}+f_{p}\tag {1}\end{equation}
\(f_{h}\) is the homogenous solution to Laplace \(\frac {\partial ^{2}f}{\partial x^{2}}+\frac {\partial ^{2}f}{\partial y^{2}}=0\) and \(f_{p}\) is a particular solution. The homogeneous solution is easily found as follows. Let
\[ f_{h}=F\left ( mx+y\right ) \]
Then \(\frac {\partial f_{h}}{\partial x}=mF^{\prime }\) and \(\frac {\partial ^{2}f_{h}}{\partial x^{2}}=m^{2}F^{\prime \prime }\) and \(\frac {\partial f_{h}}{\partial y}=F^{\prime }\) and \(\frac {\partial ^{2}f_{h}}{\partial y^{2}}=F^{\prime \prime }\) . Substituting these into \(\frac {\partial ^{2}f_{h}}{\partial x^{2}}+\frac {\partial ^{2}f_{h}}{\partial y^{2}}=0\) gives
\begin{align*} m^{2}F^{\prime \prime }+F^{\prime \prime } & =0\\ m^{2}+1 & =0\\ m & =\pm i \end{align*}
Since we assumed \(f_{h}=F\left ( mx+y\right ) \) , then the homogeneous solution is sum of two arbitrary functions (one for each root of \(m\) )
\[ f_{h}=F_{1}\left ( ix+y\right ) +F_{2}\left ( -ix+y\right ) \]
To find particular solution, let
\begin{equation} f_{p}=Ax^{n}y^{m}\tag {2}\end{equation}
\(\frac {\partial f_{p}}{\partial x}=Anx^{n-1}y^{m}\) and \(\frac {\partial ^{2}f_{p}}{\partial x^{2}}=An\left ( n-1\right ) x^{n-2}y^{m}\) and \(\frac {\partial f_{p}}{\partial y}=Ax^{n}my^{m-1}\) and \(\frac {\partial ^{2}f_{p}}{\partial y^{2}}=Ax^{n}m\left ( m-1\right ) y^{m-2}\) . Substituting these into \(\frac {\partial ^{2}f_{p}}{\partial x^{2}}+\frac {\partial ^{2}f_{p}}{\partial y^{2}}=6y\) gives
\begin{align*} An\left ( n-1\right ) x^{n-2}y^{m}+Ax^{n}m\left ( m-1\right ) y^{m-2} & =6y\\ y^{m}\left ( An\left ( n-1\right ) x^{n-2}+Ax^{n}m\left ( m-1\right ) y^{-2}\right ) & =6y \end{align*}
Comparing terms, then \(m=1\) and
\[ An\left ( n-1\right ) x^{n-2}+Ax^{n}m\left ( m-1\right ) y^{-2}=6 \]
Since \(m=1\) then the above simplifies to
\[ An\left ( n-1\right ) x^{n-2}=6 \]
Since there is no \(x\) in RHS, then \(n-2=0\) or \(n=2\) and the above becomes
\begin{align*} 2A\left ( 2-1\right ) & =6\\ A & =3 \end{align*}
Hence (2) becomes
\[ f_{p}=Ax^{n}y^{m}=3x^{2}y \]
And the complete solution (1) is
\begin{align*} f\left ( x,y\right ) & =f_{h}+f_{p}\\ & =F_{1}\left ( ix+y\right ) +F_{2}\left ( -ix+y\right ) +3x^{2}y \end{align*}