22 Wave PDE 1D infinite domain

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

problem number 153

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 ✓

ClearAll[u, t, x, c]; 
 ode = D[u[x, t], {t, 2}] + D[u[x, t], x, t] == c^2*D[u[x, t], {x, 2}]; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[ode, u[x, t], {x, t}], 60*10]];
 

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

Maple ✓

 
x:='x'; t:='t';c:='c';u:='u'; 
interface(showassumed=0); 
pde:=diff(u(x,t),t$2)+diff(u(x,t),t,x)=c^2*diff(u(x,t),x$2); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t)) assuming t>0,x>0),output='realtime'));
 

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

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22.2 With initial conditions specified, no source

problem number 154

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 ✓

ClearAll[u, t, x]; 
 pde = D[u[x, t], {t, 2}] == D[u[x, t], {x, 2}]; 
 ic = {u[x, 0] == E^(-x^2), Derivative[0, 1][u][x, 0] == 1}; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, t], {x, t}], 60*10]];
 

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

Maple ✓

 
x:='x'; t:='t'; u:='u'; 
pde := diff(u(x,t), t$2) = diff(u(x,t), x$2); 
ic:= u(x, 0) = exp(-x^2), (D[2](u))(x,0) = 1; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic], u(x, t))),output='realtime'));
 

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

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

problem number 155

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 ✓

ClearAll[u, t, x]; 
 pde = {D[u[x, t], {t, 2}] == D[u[x, t], {x, 2}] + m}; 
 ic = {u[x, 0] == Sin[x] - Cos[3*x]/E^(Abs[x]/6), Derivative[0, 1][u][x, 0] == 0}; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, t], {x, t}], 60*10]];
 

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

Maple ✓

 
x:='x'; t:='t'; u:='u'; 
pde:= diff(u(x, t), t$2) = diff(u(x, t), x$2) + m; 
ic := u(x, 0) = sin(x) - cos(3*x)/exp(abs(x)/6), (D[2](u))(x, 0) =0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic], u(x, t))),output='realtime'));
 

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

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22.4 non-linear wave PDE (Solitons)

problem number 156

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 ✓

ClearAll[u, t, x]; 
 pde = D[u[x, t], t] + 6*u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 3}] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

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

Maple ✓

 
x:='x'; t:='t'; u:='u'; 
pde :=  diff(u(x,t),t)+6*u(x,t)*diff(u(x,t),x)+diff(u(x,t),x$3)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t)) assuming t>0,x>0),output='realtime'));
 

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

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22.5 Hyperbolic PDE with non-rational coefficients

problem number 157

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 ✓

ClearAll[u, x, y]; 
 ode = D[u[x, y], {x, 2}] - 2*Sin[x]*D[u[x, y], x, y] - Cos[x]^2*D[u[x, y], {y, 2}] - Cos[x]*D[u[x, y], y] == 0; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[ode, u[x, y], {x, y}], 60*10]];
 

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

Maple ✗

 
x:='x'; t:='t';c:='c';u:='u'; 
interface(showassumed=0); 
ode := diff(u(x, y), x$2) - 2*sin(x)*diff(u(x, y),x,y)-cos(x)^2*diff(u(x, y), y$2) - cos(x)*diff(u(x, y), y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(ode, u(x, y))),output='realtime'));
 

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

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22.6 Inhomogeneous hyperbolic PDE with constant coefficients

problem number 158

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 ✓

ClearAll[u, x, t]; 
 ode = 3*D[u[x, t], {x, 2}] - D[u[x, t], {t, 2}] + D[u[x, t], x, t] == 1; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[ode, u[x, t], {x, t}], 60*10]];
 

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

Maple ✓

 
x:='x'; t:='t';y:='y';u:='u'; 
ode := 3*diff(u(x, t), x$2) - diff(u(x, t),t$2)+diff(u(x, t), x,t) =1; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(ode, u(x, t))),output='realtime'));
 

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

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22.7 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

problem number 159

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 ✓

ClearAll[u, v, x, t]; 
 eqns = {D[u[x, t], t] == D[v[x, t], x] + 1, D[v[x, t], t] == -D[u[x, t], x] - 1}; 
 ic = {u[x, 0] == Cos[x]^2, v[x, 0] == Sin[x]}; 
 sol = AbsoluteTiming[TimeConstrained[FullSimplify[DSolve[{eqns, ic}, {u[x, t], v[x, t]}, {x, t}]], 60*10]];
 

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

Maple ✗

 
x:='x'; t:='t';v:='v';u:='u'; 
pde1 := diff(u(x, t), t) = diff(v(x, t), x) + 1; 
pde2 := diff(v(x, t), t) = -diff(u(x, t), x) - 1; 
ic := u(x, 0) = cos(x)^2, v(x, 0) = sin(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde1,pde2, ic], {u(x, t), v(x, t)})),output='realtime'));
 

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