7 Heat PDE on infinite domain, 1D

 7.1 No boundary conditions specified. Initial conditions \(u(x,0)=\sin (x)\) and no source
 7.2 No intial conditions and no boundary conditions specified
 7.3 with source and a piecewise initial condition at \(t>0\)

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7.1 No boundary conditions specified. Initial conditions \(u(x,0)=\sin (x)\) and no source

problem number 69

Solve the heat equation

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

For \(-\infty <x<\infty \) and \(t>0\). The boundary conditions are

Initial condition is \(u(x,0)=\sin (x)\)

Mathematica

\[ \left \{\left \{u(x,t)\to e^{-k t} \sin (x)+m t\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =\sin \left ( x \right ){{\rm e}^{-k\,t}}+mt \]

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7.2 No intial conditions and no boundary conditions specified

problem number 70

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

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

Mathematica

\[ \text{DSolve}\left [u^{(0,1)}(x,t)=u^{(2,0)}(x,t),u(x,t),\{x,t\}\right ] \]

Maple

\[ u \left ( x,t \right ) ={\it \_C3}\,{{\rm e}^{{\it \_c}_{{1}}t}}{\it \_C1}\,{{\rm e}^{\sqrt{{\it \_c}_{{1}}}x}}+{\frac{{\it \_C3}\,{{\rm e}^{{\it \_c}_{{1}}t}}{\it \_C2}}{{{\rm e}^{\sqrt{{\it \_c}_{{1}}}x}}}} \]

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7.3 with source and a piecewise initial condition at \(t>0\)

problem number 71

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 \(u(x,t)\) on real line with \(t>0\)

\[ \frac{ \partial u}{\partial t}+ 1 = \mu \frac{ \partial ^2 u}{\partial x^2} \] With initial condition \[ u(x,1)= \begin{cases} 0 & x \geq 0 \\ 1 & x < 0 \end{cases} \]

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)+1=\mu u^{(2,0)}(x,t),u(x,1)=\left (\begin{array}{cc} \{ & \begin{array}{cc} 1 & x\leq 0 \\ 0 & \text{True} \\\end{array} \\\end{array}\right )\right \},u(x,t),x,t,\text{Assumptions}\to \mu >0\right ] \] due to i.c. not at zero

Maple

\[ u \left ( x,t \right ) =3/2-1/2\,\erf \left ( 1/2\,{\frac{x}{\sqrt{t-1}\sqrt{\mu }}} \right ) -t \]