### 18 Wave PDE on semi-inﬁnite domain

_______________________________________________________________________________________

#### 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

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

_______________________________________________________________________________________

#### 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

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

_______________________________________________________________________________________

#### 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

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

_______________________________________________________________________________________

#### 18.4 With initial conditions given at $$t=1$$ instead of $$t=0$$

problem number 133

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

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

_______________________________________________________________________________________

#### 18.5 initial conditions at $$t=0$$ but B.C. at $$x=1$$

problem number 134

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

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

_______________________________________________________________________________________

#### 18.6 initial conditions at $$t=1$$ with source that depends on time and space

problem number 135

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

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

_______________________________________________________________________________________

#### 18.7 Left end free with initial position and velocity given

problem number 136

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

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