6 Heat PDE on semi-infinite domain (1D)

 6.1 Left end \(u(0,t)=f(t)\), zero initial conditions, No source (Logan p. 76)
 6.2 Left end \(u(0,t)=1\), zero initial conditions, No source
 6.3 B.C. not located at \(x=0\) and I.C. not at \(t=0\)
 6.4 Left end at \(u(0,t)=\mu \), non zero initial conditions, No source
 6.5 No condition on left end. Initial conditions given, no source
 6.6 With advection term and initial conditons
 6.7 Initial position as a UnitBox function
 6.8 Left end fixed at specific temperature, non-zero initial conditions, no source
 6.9 Left end function of temperature, zero initial conditions, no source
 6.10 Left end insulated, initial conditions triangle function. No source
 6.11 one end insulated at arbitray location, initial conditions not at \(t=0\). No source

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6.1 Left end \(u(0,t)=f(t)\), zero initial conditions, No source (Logan p. 76)

problem number 58

This is problem at page 76 from David J Logan text book.

Solve the heat equation

\[ \frac{ \partial u}{\partial t} = \frac{ \partial ^2 u}{\partial x^2} \]

The boundary conditions are \(u(0,t)=f(t)\) and initial conditions \(u(x,0)=0\)

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{x \int _0^t \frac{f(z) e^{-\frac{x^2}{4 (t-z)}}}{(t-z)^{3/2}} \, dz}{2 \sqrt{\pi }}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =1/2\,{\frac{x}{\sqrt{\pi }}\int _{0}^{t}\!{\frac{f \left ( \zeta \right ) }{ \left ( t-\zeta \right ) ^{3/2}}{{\rm e}^{-{\frac{{x}^{2}}{4\,t-4\,\zeta }}}}}\,{\rm d}\zeta } \]

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6.2 Left end \(u(0,t)=1\), zero initial conditions, No source

problem number 59

Solve the heat equation

\[ \frac{ \partial u}{\partial t} = k \frac{ \partial ^2 u}{\partial x^2} \]

For \(x>0\) and \(t>0\). The boundary conditions is \(u(0,t)=1\) and And initial condition \(u(x,0)=0\)

Mathematica

\[ \left \{\left \{u(x,t)\to \text{Erfc}\left (\frac{x}{2 \sqrt{k t}}\right )\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =1-\erf \left ( 1/2\,{\frac{x}{\sqrt{t}\sqrt{k}}} \right ) \]

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6.3 B.C. not located at \(x=0\) and I.C. not at \(t=0\)

problem number 60

Added December 20, 2018.

From https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Solve the heat equation for \(u(x,t)\)

\[ \frac{ \partial u}{\partial t}= \frac{1}{4} \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,t_0)= 10; \] And boundary conditions \[ u(-x_0,t) = 0 \] For \(x>|x_0|\) and \(t>|t_0|\).

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)=\frac{1}{4} u^{(2,0)}(x,t),u(-\text{x0},t)=0,u(x,\text{t0})=10\right \},u(x,t),x,t,\text{Assumptions}\to \{t>\left | \text{t0}\right | ,x>\left | \text{x0}\right | \}\right ] \] due to IC/BC not zero

Maple

\[ u \left ( x,t \right ) =10\,\erf \left ({\frac{x+{\it x0}}{\sqrt{t-{\it t0}}}} \right ) \]

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6.4 Left end at \(u(0,t)=\mu \), non zero initial conditions, No source

problem number 61

Solve the heat equation

\[ \frac{ \partial u}{\partial t} = k \frac{ \partial ^2 u}{\partial x^2} \]

For \(x>0\) and \(t>0\). The boundary conditions is \(u(0,t)=\mu \) and And initial condition \(u(x,0)=\lambda \)

Mathematica

\[ \left \{\left \{u(x,t)\to \mu \text{Erf}\left (\frac{x}{2 \sqrt{k t}}\right )+\lambda \text{Erfc}\left (\frac{x}{2 \sqrt{k t}}\right )\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) = \left ( -\lambda +\mu \right ) \erf \left ( 1/2\,{\frac{x}{\sqrt{t}\sqrt{k}}} \right ) +\lambda \]

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6.5 No condition on left end. Initial conditions given, no source

problem number 62

From Mathematica DSolve help pages. Solve the heat equation for \(u(x,t)\) on real line with \(t>0\)

\[ \frac{ \partial u}{\partial t}= \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,0)= e^{-x^2} \]

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{e^{-\frac{x^2}{4 t+1}}}{\sqrt{4 t+1}}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) ={\frac{1}{\sqrt{1+4\,t}}{{\rm e}^{-{\frac{{x}^{2}}{1+4\,t}}}}} \]

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6.6 With advection term and initial conditons

problem number 63

From Mathematica DSolve help pages. Solve the heat equation for \(u(x,t)\) on real line with \(t>0\)

\[ \frac{ \partial u}{\partial t}= 12 \frac{ \partial ^2 u}{\partial x^2} + \sin t \frac{\partial u}{\partial x} \] With initial condition \[ u(x,0)= x \]

Mathematica

\[ \{\{u(x,t)\to -\cos (t)+x+1\}\} \]

Maple

\[ u \left ( x,t \right ) =-\cos \left ( t \right ) +x+1 \]

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6.7 Initial position as a UnitBox function

problem number 64

From Mathematica DSolve help pages. Solve the heat equation for \(u(x,t)\) on real line with \(t>0\)

\[ \frac{ \partial u}{\partial t}= \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,0)= \text{UnitBox[x]} \] Where UnitBox is equal to 1 if \(|x| \leq \frac{1}{2}\) and zero otherwise.

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{1}{2} \left (\text{Erf}\left (\frac{1-2 x}{4 \sqrt{t}}\right )+\text{Erf}\left (\frac{2 x+1}{4 \sqrt{t}}\right )\right )\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-1/2\,\erf \left ( 1/4\,{\frac{2\,x-1}{\sqrt{t}}} \right ) +1/2\,\erf \left ( 1/4\,{\frac{2\,x+1}{\sqrt{t}}} \right ) \]

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6.8 Left end fixed at specific temperature, non-zero initial conditions, no source

problem number 65

From Mathematica DSolve help pages.

Solve the heat equation for \(u(x,t)\) on half the line \(x>0\) and \(t>0\)

\[ \frac{ \partial u}{\partial t}= \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,0)= \cos x \] And boundary conditions \[ u(0,t)= 1 \]

Mathematica

\[ \left \{\left \{u(x,t)\to \begin{array}{cc} \{ & \begin{array}{cc} \frac{i e^{-\frac{x^2}{4 t}} \left (\text{DawsonF}\left (\frac{2 t-i x}{2 \sqrt{t}}\right )-\text{DawsonF}\left (\frac{2 t+i x}{2 \sqrt{t}}\right )\right )}{\sqrt{\pi }}+\text{Erfc}\left (\frac{x}{2 \sqrt{t}}\right ) & x>0 \\ \text{Indeterminate} & \text{True} \\\end{array} \\\end{array}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =1/2\,{{\rm e}^{-t+ix}}\erf \left ( 1/2\,{\frac{2\,it+x}{\sqrt{t}}} \right ) -\erf \left ( 1/2\,{\frac{x}{\sqrt{t}}} \right ) -1/2\,{{\rm e}^{-t-ix}}\erf \left ( 1/2\,{\frac{2\,it-x}{\sqrt{t}}} \right ) +1 \]

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6.9 Left end function of temperature, zero initial conditions, no source

problem number 66

Solve the heat equation for \(u(x,t)\) on half the line \(x>0\) and \(t>0\)

\[ \frac{ \partial u}{\partial t}= k \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,0)=0 \] And boundary conditions \begin{align*} u(0,t) &= \sin (t) u(\infty ,t) &\leq \infty \end{align*}

The last condition above means it is bounded at infinity.

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{\left (2 k t+x^2\right ) \text{Erfc}\left (\frac{x}{2 \sqrt{k t}}\right )-\frac{2 x \sqrt{k t} e^{-\frac{x^2}{4 k t}}}{\sqrt{\pi }}}{2 k}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-{\frac{1}{k\,\sqrt{\pi }} \left ({{\rm e}^{-1/4\,{\frac{{x}^{2}}{k\,t}}}}\sqrt{k}\sqrt{t}x+ \left ( k\,t+1/2\,{x}^{2} \right ) \left ( \erf \left ( 1/2\,{\frac{x}{\sqrt{k}\sqrt{t}}} \right ) -1 \right ) \sqrt{\pi } \right ) } \]

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6.10 Left end insulated, initial conditions triangle function. No source

problem number 67

From Mathematica DSolve help pages.

Solve the heat equation for \(u(x,t)\) on half the line \(x>0\) and \(t>0\)

\[ \frac{ \partial u}{\partial t}= \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,0)= \text{UnitTriagle[x-3]} \] And boundary conditions \[ \frac{ \partial u}{\partial x}(0,t)= 0 \]

Mathematica

\[ \left \{\left \{u(x,t)\to \begin{array}{cc} \{ & \begin{array}{cc} \frac{1}{2} \left (\frac{\text{Erf}\left (\frac{\left | x-4\right | }{2 \sqrt{t}}\right ) (x-4)^2}{\left | 4-x\right | }+(x+2) \text{Erf}\left (\frac{x+2}{2 \sqrt{t}}\right )-2 (x+3) \text{Erf}\left (\frac{x+3}{2 \sqrt{t}}\right )+(x+4) \text{Erf}\left (\frac{x+4}{2 \sqrt{t}}\right )-\frac{2 (x-3)^2 \text{Erf}\left (\frac{\left | x-3\right | }{2 \sqrt{t}}\right )}{\left | 3-x\right | }+\frac{(x-2)^2 \text{Erf}\left (\frac{\left | x-2\right | }{2 \sqrt{t}}\right )}{\left | 2-x\right | }+\frac{2 \left (e^{-\frac{(x-4)^2}{4 t}}-2 e^{-\frac{(x-3)^2}{4 t}}+e^{-\frac{(x-2)^2}{4 t}}+e^{-\frac{(x+2)^2}{4 t}}-2 e^{-\frac{(x+3)^2}{4 t}}+e^{-\frac{(x+4)^2}{4 t}}\right ) \sqrt{t}}{\sqrt{\pi }}\right ) & x>0 \\ \text{Indeterminate} & \text{True} \\\end{array} \\\end{array}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) ={\frac{1}{\sqrt{\pi }\sqrt{t}} \left ( t{{\rm e}^{-1/4\,{\frac{ \left ( -4+x \right ) ^{2}}{t}}}}-2\,t{{\rm e}^{-1/4\,{\frac{ \left ( -3+x \right ) ^{2}}{t}}}}+t{{\rm e}^{-1/4\,{\frac{ \left ( -2+x \right ) ^{2}}{t}}}}+t{{\rm e}^{-1/4\,{\frac{ \left ( 2+x \right ) ^{2}}{t}}}}+t{{\rm e}^{-1/4\,{\frac{ \left ( 4+x \right ) ^{2}}{t}}}}-2\,t{{\rm e}^{-1/4\,{\frac{ \left ( x+3 \right ) ^{2}}{t}}}}+1/2\, \left ( \left ( -4+x \right ) \erf \left ( 1/2\,{\frac{-4+x}{\sqrt{t}}} \right ) + \left ( -2\,x+6 \right ) \erf \left ( 1/2\,{\frac{-3+x}{\sqrt{t}}} \right ) + \left ( -2+x \right ) \erf \left ( 1/2\,{\frac{-2+x}{\sqrt{t}}} \right ) + \left ( 2+x \right ) \erf \left ( 1/2\,{\frac{2+x}{\sqrt{t}}} \right ) + \left ( 4+x \right ) \erf \left ( 1/2\,{\frac{4+x}{\sqrt{t}}} \right ) -2\,\erf \left ( 1/2\,{\frac{x+3}{\sqrt{t}}} \right ) \left ( x+3 \right ) \right ) \sqrt{\pi }\sqrt{t} \right ) } \]

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6.11 one end insulated at arbitray location, initial conditions not at \(t=0\). No source

problem number 68

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Solve for \(u(x,t)\) for \(t>0,x>0\)

\[ \frac{ \partial u}{\partial t}= \frac{1}{4} \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,t_0)= 10 e^{-x^2} \] And boundary conditions \[ \frac{ \partial u}{\partial x}(x_0,t)= 0 \]

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)=\frac{1}{4} u^{(2,0)}(x,t),u(x,\text{t0})=10 e^{-x^2},u^{(1,0)}(\text{x0},t)=0\right \},u(x,t),\{x,t\},\text{Assumptions}\to \{x>0,t>0\}\right ] \]

Maple

\[ u \left ( x,t \right ) =5\,{\frac{1}{\sqrt{t-{\it t0}+1}} \left ({{\rm e}^{4\,{\frac{{\it x0}\, \left ( -x+{\it x0} \right ) }{-t+{\it t0}-1}}}}\erf \left ({\frac{ \left ({\it t0}-t+1 \right ){\it x0}-x}{\sqrt{t-{\it t0}+1}\sqrt{t-{\it t0}}}} \right ) +{{\rm e}^{4\,{\frac{{\it x0}\, \left ( -x+{\it x0} \right ) }{-t+{\it t0}-1}}}}+\erf \left ({\frac{ \left ( -t+{\it t0}-1 \right ){\it x0}+x}{\sqrt{t-{\it t0}+1}\sqrt{t-{\it t0}}}} \right ) +1 \right ){{\rm e}^{{\frac{{x}^{2}}{-t+{\it t0}-1}}}}} \]