18 Wave PDE on semi-infinite domain

 18.1 Left end having a moving boundary condition
 18.2 Initial value with a Dirichlet condition on the half-line
 18.3 Initial value problem with a Neumann condition on the half-line
 18.4 With initial conditions given at \(t=1\) instead of \(t=0\)
 18.5 initial conditions at \(t=0\) but B.C. at \(x=1\)
 18.6 initial conditions at \(t=1\) with source that depends on time and space
 18.7 Left end free with initial position and velocity given

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18.1 Left end having a moving boundary condition

problem number 130

Solve for \(u(x,t)\) with \(t>0\) and \(x>0\) \[ \frac{\partial ^2 u}{\partial t^2} = c^2 \frac{\partial ^2 u}{\partial x^2} \]

With boundary conditions

\begin{align*} u(0,t) &= g(t) \end{align*}

With initial conditions

\begin{align*} \frac{\partial u}{\partial t}(x,0) &=0 \\ u(x,0) &= 0 \end{align*}

Mathematica

\[ \left \{\left \{u(x,t)\to \begin{array}{cc} \{ & \begin{array}{cc} 0 & x>c t \\ g\left (t-\frac{x}{c}\right ) & x\leq c t \\ \text{Indeterminate} & \text{True} \\\end{array} \\\end{array}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-{\it invlaplace} \left ({{\rm e}^{-{\frac{sx}{c}}}}{\it \_F2} \left ( s \right ) ,s,t \right ) +{\it Heaviside} \left ({\frac{tc-x}{c}} \right ) g \left ({\frac{tc-x}{c}} \right ) +{\it invlaplace} \left ({\it \_F2} \left ( s \right ){{\rm e}^{{\frac{sx}{c}}}},s,t \right ) \]

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18.2 Initial value with a Dirichlet condition on the half-line

problem number 131

Taken from Mathematica DSolve help pages.

Solve for \(u(x,t)\) initial value wave PDE on infinite domain with \(t>0\) and \(x>0\).

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

With initial conditions

\begin{align*} u(x,0) &= \sin ^2(x) \hspace{20pt} \pi <x< 2\pi \\ \frac{\partial u}{\partial t}(x,0) &= 0 \end{align*}

And boundary conditions \(u(0,t)=0\)

Mathematica

\[ \left \{\left \{u(x,t)\to \begin{array}{cc} \{ & \begin{array}{cc} \frac{1}{2} \left (\left (\begin{array}{cc} \{ & \begin{array}{cc} \sin ^2\left (\sqrt{c^2} t-x\right ) & \pi <x-\sqrt{c^2} t<2 \pi \\ 0 & \text{True} \\\end{array} \\\end{array}\right )+\left (\begin{array}{cc} \{ & \begin{array}{cc} \sin ^2\left (\sqrt{c^2} t+x\right ) & \pi <\sqrt{c^2} t+x<2 \pi \\ 0 & \text{True} \\\end{array} \\\end{array}\right )\right ) & x>\sqrt{c^2} t\geq 0 \\ \frac{1}{2} \left (\left (\begin{array}{cc} \{ & \begin{array}{cc} \sin ^2\left (\sqrt{c^2} t+x\right ) & \pi <\sqrt{c^2} t+x<2 \pi \\ 0 & \text{True} \\\end{array} \\\end{array}\right )-\left (\begin{array}{cc} \{ & \begin{array}{cc} \sin ^2\left (\sqrt{c^2} t-x\right ) & \pi <\sqrt{c^2} t-x<2 \pi \\ 0 & \text{True} \\\end{array} \\\end{array}\right )\right ) & 0\leq x\leq \sqrt{c^2} t \\ \text{Indeterminate} & \text{True} \\\end{array} \\\end{array}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =\cases{\cases{0&$tc+x\leq \pi $\cr 1/2\, \left ( \sin \left ( tc+x \right ) \right ) ^{2}&$tc+x<2\,\pi $\cr 0&$2\,\pi \leq tc+x$\cr }+\cases{0&$tc-x\leq \pi $\cr -1/2\, \left ( \sin \left ( tc-x \right ) \right ) ^{2}&$tc-x<2\,\pi $\cr 0&$2\,\pi \leq tc-x$\cr }&$x<tc$\cr \cases{0&$tc+x\leq \pi $\cr 1/2\, \left ( \sin \left ( tc+x \right ) \right ) ^{2}&$tc+x<2\,\pi $\cr 0&$2\,\pi \leq tc+x$\cr }+\cases{0&$-tc+x\leq \pi $\cr 1/2\, \left ( \sin \left ( tc-x \right ) \right ) ^{2}&$-tc+x<2\,\pi $\cr 0&$2\,\pi \leq -tc+x$\cr }&$tc<x$\cr } \]

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18.3 Initial value problem with a Neumann condition on the half-line

problem number 132

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

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

With initial conditions

\begin{align*} u(x,0) &= \sin ^3(x) \\ \frac{\partial u}{\partial t}(x,0) &= 1-e^{- \frac{x}{10}} \end{align*}

And boundary conditions \(\frac{\partial u}{\partial x}(0,t)=1\)

Mathematica

\[ \begin{cases} \frac{1}{2} \left (\sin ^3\left (\sqrt{c^2} t+x\right )-\sin ^3\left (\sqrt{c^2} t-x\right )\right )+\frac{2 \sqrt{c^2} t-20 e^{-x/10} \sinh \left (\frac{\sqrt{c^2} t}{10}\right )}{2 \sqrt{c^2}} & x>\sqrt{c^2} t\geq 0 \\ \frac{10 e^{\frac{1}{10} \left (-\sqrt{c^2} t-x\right )}+10 e^{\frac{1}{10} \left (x-\sqrt{c^2} t\right )}+2 \sqrt{c^2} t-20}{2 \sqrt{c^2}}-\sqrt{c^2} \left (t-\frac{x}{\sqrt{c^2}}\right )+\frac{1}{2} \left (\sin ^3\left (\sqrt{c^2} t-x\right )+\sin ^3\left (\sqrt{c^2} t+x\right )\right ) & 0\leq x\leq \sqrt{c^2} t\end{cases} \]

Maple

\[ u \left ( x,t \right ) =\cases{1/2\,{\frac{ \left ( \sin \left ( tc+x \right ) \right ) ^{3}c- \left ( \sin \left ( tc-x \right ) \right ) ^{3}c+2\,tc-10\,{{\rm e}^{1/10\,tc-x/10}}+10\,{{\rm e}^{-1/10\,tc-x/10}}}{c}}&$tc<x$\cr 1/2\,{\frac{10\,{{\rm e}^{-1/10\,tc-x/10}}+10\,{{\rm e}^{-1/10\,tc+x/10}}+ \left ( \sin \left ( tc-x \right ) \right ) ^{3}c+ \left ( \sin \left ( tc+x \right ) \right ) ^{3}c-20-2\,{c}^{2}t+ \left ( 2\,t+2\,x \right ) c}{c}}&$x<tc$\cr } \]

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18.4 With initial conditions given at \(t=1\) instead of \(t=0\)

problem number 133

Added December 20, 2018.

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

Solve

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

With initial conditions

\begin{align*} u(x,1) &= e^{-(x-6)^2}+e^{-(x+6)^2} \\ \frac{\partial u}{\partial t}(x,1) &= \frac{1}{2} \end{align*}

Mathematica

\[ \begin{cases} \frac{1}{2} \left (\sin ^3\left (\sqrt{c^2} t+x\right )-\sin ^3\left (\sqrt{c^2} t-x\right )\right )+\frac{2 \sqrt{c^2} t-20 e^{-x/10} \sinh \left (\frac{\sqrt{c^2} t}{10}\right )}{2 \sqrt{c^2}} & x>\sqrt{c^2} t\geq 0 \\ \frac{10 e^{\frac{1}{10} \left (-\sqrt{c^2} t-x\right )}+10 e^{\frac{1}{10} \left (x-\sqrt{c^2} t\right )}+2 \sqrt{c^2} t-20}{2 \sqrt{c^2}}-\sqrt{c^2} \left (t-\frac{x}{\sqrt{c^2}}\right )+\frac{1}{2} \left (\sin ^3\left (\sqrt{c^2} t-x\right )+\sin ^3\left (\sqrt{c^2} t+x\right )\right ) & 0\leq x\leq \sqrt{c^2} t\end{cases} \]

Maple

\[ u \left ( x,t \right ) =1/2\,{{\rm e}^{- \left ( -x+t+5 \right ) ^{2}}}+1/2\,{{\rm e}^{- \left ( -x+t-7 \right ) ^{2}}}+1/2\,{{\rm e}^{- \left ( x+t-7 \right ) ^{2}}}+1/2\,{{\rm e}^{- \left ( x+t+5 \right ) ^{2}}}+t/2-1/2 \]

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18.5 initial conditions at \(t=0\) but B.C. at \(x=1\)

problem number 134

Added December 20, 2018.

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

Solve

\[ \frac{\partial ^2 u}{\partial t^2} = \frac{1}{4} \frac{\partial ^2 u}{\partial x^2} \]

With initial conditions

\begin{align*} u(x,0) &= e^{-x^2}\\ \frac{\partial u}{\partial t}(x,0) &= 0 \end{align*}

And Boundary conditions \(\frac{\partial u}{\partial x}(1,t)= 0\)

Mathematica

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

Maple

\[ u \left ( x,t \right ) =\cases{1/2\,{{\rm e}^{-1/4\, \left ( t+2\,x \right ) ^{2}}}+1/2\,{{\rm e}^{-1/4\, \left ( t-2\,x \right ) ^{2}}}&$t/2<x-1$\cr 1/2\,{{\rm e}^{-1/4\, \left ( t+2\,x \right ) ^{2}}}+1/2\,{{\rm e}^{-1/4\, \left ( 4+t-2\,x \right ) ^{2}}}&$x-1<t/2$\cr } \]

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18.6 initial conditions at \(t=1\) with source that depends on time and space

problem number 135

Added December 20, 2018.

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

Solve

\[ \frac{\partial ^2 u}{\partial t^2} = c^2 \frac{\partial ^2 u}{\partial x^2} + f(x,t) \]

With initial conditions

\begin{align*} u(x,1) &= g(x)\\ \frac{\partial u}{\partial t}(x,1) &= h(x) \end{align*}

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,2)}(x,t)=c^2 u^{(2,0)}(x,t),\left \{u(x,1)=g(x),u^{(0,1)}(x,1)=h(x)\right \}\right \},u(x,t),\{x,t\},\text{Assumptions}\to \{t>0,x>0\}\right ] \]

Maple

\[ u \left ( x,t \right ) =1/2\,{\frac{\int _{0}^{t-1}\!\int _{ \left ( -t+\tau +1 \right ) c+x}^{x+c \left ( t-1-\tau \right ) }\!{c}^{2} \left ({\frac{{\rm d}^{2}}{{\rm d}{\zeta }^{2}}}h \left ( \zeta \right ) \right ) \tau +{c}^{2}{\frac{{\rm d}^{2}}{{\rm d}{\zeta }^{2}}}g \left ( \zeta \right ) +f \left ( \zeta ,\tau +1 \right ) \,{\rm d}\zeta \,{\rm d}\tau + \left ( 2\,t-2 \right ) ch \left ( x \right ) +2\,g \left ( x \right ) c}{c}} \]

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18.7 Left end free with initial position and velocity given

problem number 136

Added December 20, 2018.

Example 17, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

\[ \frac{\partial ^2 u}{\partial t^2} = 9 \frac{\partial ^2 u}{\partial x^2} + f(x,t) \]

With initial conditions

\begin{align*} u(x,0) &= 0\\ \frac{\partial u}{\partial t}(x,0) &= x^3 \end{align*}

And boundary condition \(\frac{\partial u}{\partial x}(0,t) = 0\).

Mathematica

\[ \left \{\left \{u(x,t)\to 3 c_1 \delta (3 t-x)+\frac{1}{12} (x-3 t)^4 \theta \left (t-\frac{x}{3}\right )+9 t^3 x+t x^3\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =\cases{9\,{t}^{3}x+t{x}^{3}&$3\,t<x$\cr{\frac{27\,{t}^{4}}{4}}+9/2\,{t}^{2}{x}^{2}+1/12\,{x}^{4}&$x<3\,t$\cr } \]