19 Wave PDE 1D infinite domain

 19.1 General solution for a second-order hyperbolic PDE on real line
 19.2 With initial conditions specified, no source
 19.3 Wave PDE on infinite domain with initial conditions specified, with source term
 19.4 non-linear wave PDE (Solitons)
 19.5 Hyperbolic PDE with non-rational coefficients
 19.6 Inhomogeneous hyperbolic PDE with constant coefficients
 19.7 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

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19.1 General solution for a second-order hyperbolic PDE on real line

problem number 137

From Mathematica DSolve help pages (slightly modified)

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

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

Mathematica

\[ \left \{\left \{u(x,t)\to c_1\left (t-\frac{\left (\sqrt{4 c^2+1}-1\right ) x}{2 c^2}\right )+c_2\left (t-\frac{\left (-\sqrt{4 c^2+1}-1\right ) x}{2 c^2}\right )\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) ={\it \_F1} \left ( 1/2\,{\frac{2\,{c}^{2}t+x\sqrt{4\,{c}^{2}+1}+x}{{c}^{2}}} \right ) +{\it \_F2} \left ( 1/2\,{\frac{2\,{c}^{2}t-x\sqrt{4\,{c}^{2}+1}+x}{{c}^{2}}} \right ) \]

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19.2 With initial conditions specified, no source

problem number 138

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

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

With initial conditions

\begin{align*} u(x,0) &=e^{-x^2} \\ \frac{\partial u}{\partial t}(x,0) &= 1 \end{align*}

Mathematica

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

Maple

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

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19.3 Wave PDE on infinite domain with initial conditions specified, with source term

problem number 139

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

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

With initial conditions

\begin{align*} u(x,0) &=\sin x- \frac{\cos 3 x}{e^{ \frac{abs(x)}{6} }} \\ \frac{\partial u}{\partial t}(x,0) &= 0 \end{align*}

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{1}{2} \left (-e^{-\frac{\left | x-t\right | }{6}} \cos (3 (x-t))-e^{-\frac{\left | t+x\right | }{6}} \cos (3 (t+x))-\sin (t-x)+\sin (t+x)\right )+\frac{m t^2}{2}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =1/2\, \left ( \left ( m{t}^{2}-\sin \left ( -x+t \right ) +\sin \left ( x+t \right ) \right ){{\rm e}^{1/6\, \left | -x+t \right | +1/6\, \left | x+t \right | }}-{{\rm e}^{1/6\, \left | -x+t \right | }}\cos \left ( 3\,x+3\,t \right ) -\cos \left ( 3\,t-3\,x \right ){{\rm e}^{1/6\, \left | x+t \right | }} \right ){{\rm e}^{-1/6\, \left | -x+t \right | -1/6\, \left | x+t \right | }} \]

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19.4 non-linear wave PDE (Solitons)

problem number 140

This was first solved analytically by (Krvskal, Zabrsky 1965).

Solve

\[ \frac{\partial u}{\partial t} +6 u(x,t) \frac{\partial u}{\partial x} + \frac{\partial ^3 u}{\partial x^3} = 0 \]

Mathematica

\[ \left \{\left \{u(x,t)\to -\frac{12 c_1^3 \tanh ^2\left (c_2 t+c_1 x+c_3\right )-8 c_1^3+c_2}{6 c_1}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/6\,{\frac{8\,{{\it \_C2}}^{3}-{\it \_C3}}{{\it \_C2}}} \] Returning a solution that is not the most general one

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19.5 Hyperbolic PDE with non-rational coefficients

problem number 141

From Mathematica DSolve help pages

Solve for \(u(x,y)\)

\[ \frac{\partial ^2 u}{\partial x^2} -2 \sin x \frac{\partial ^2 u}{\partial x \partial y } -\cos ^2 x \frac{\partial ^2 u}{\partial y^2} -\cos x \frac{\partial u}{\partial y}=0 \]

Mathematica

\[ \left \{\left \{u(x,y)\to c_1(x-\cos (x)+y)+c_2(-x-\cos (x)+y)\right \}\right \} \]

Maple

\[ \text{ sol=() } \]

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19.6 Inhomogeneous hyperbolic PDE with constant coefficients

problem number 142

From Mathematica DSolve help pages

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

\[ 3 \frac{\partial ^2 u}{\partial x^2} - \frac{\partial ^2 u}{\partial t^2} + \frac{\partial ^2 u}{\partial x \partial t}=1 \]

Mathematica

\[ \left \{\left \{u(x,t)\to c_1\left (t-\frac{1}{6} \left (1+\sqrt{13}\right ) x\right )+c_2\left (t-\frac{1}{6} \left (1-\sqrt{13}\right ) x\right )+\frac{x^2}{6}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) ={\it \_F2} \left ( 1/6\, \left ( -1+\sqrt{13} \right ) x+t \right ) +{\it \_F1} \left ( 1/2\, \left ( 1/13\,\sqrt{13}+1 \right ) x-3/13\,t\sqrt{13} \right ) +1/13\,\sqrt{13} \left ( 1/6\, \left ( -1+\sqrt{13} \right ) x+t \right ) \left ( 1/2\, \left ( 1/13\,\sqrt{13}+1 \right ) x-3/13\,t\sqrt{13} \right ) \]

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19.7 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

problem number 143

From Mathematica DSolve help pages

Solve for \(u(x,t),v(x,t)\)

\begin{align*} \frac{\partial u}{\partial t} &= \frac{\partial v}{\partial x}+1\\ \frac{\partial v}{\partial t} &= -\frac{\partial u}{\partial x}-1 \end{align*}

With initial conditions \begin{align*} u(x,0) &= \cos ^2 x\\ v(x,0) &= \sin x \end{align*}

Mathematica

\[ \left \{\left \{u(x,t)\to \sinh (t) \cos (x)+\frac{1}{2} \cosh (2 t) \cos (2 x)+t+\frac{1}{2},v(x,t)\to \cosh (t) \sin (x) (2 \sinh (t) \cos (x)+1)-t\right \}\right \} \]

Maple

\[ \text{ sol=() } \]