____________________________________________________________________________________
Taken from Maple PDE help pages. This wave PDE inside square with free to move on left edge and right edge, and top and bottom edges are fixed. It has zero initial velocity, but given a non-zero initial position. Where \(0<x<\pi \) and \(0<y<\pi \) and \(t>0\).
Solve \[ \frac {\partial ^2 u}{\partial t^2} = \frac {1}{4} \left ( \frac {\partial ^2 u}{\partial x^2}+ \frac {\partial ^2 u}{\partial y^2} \right ) \]
With boundary conditions
\begin {align*} \frac {\partial u}{\partial x}u(0,y,t) &= 0 \\ \frac {\partial u}{\partial x}u(\pi ,y,t) &= 0 \\ u(x,0,t) &= 0\\ u(x,\pi ,0) &=0 \end {align*}
With initial conditions
\begin {align*} \frac {\partial u}{\partial t}(x,y,0) &=0 \\ u(x,0) &= x y (\pi -y) \end {align*}
Mathematica ✗
ClearAll[u, t, y, x]; pde = D[u[x, y, t], {t, 2}] == (1*(D[u[x, y, t], {x, 2}] + D[u[x, y, t], {y, 2}]))/4; ic = {Derivative[0, 0, 1][u][x, y, 0] == 0, u[x, y, 0] == x*y*(Pi - y)}; bc = {Derivative[1, 0, 0][u][0, y, t] == 0, Derivative[1, 0, 0][u][Pi, y, t] == 0, u[x, 0, t] == 0, u[x, Pi, t] == 0}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, bc, ic}, u[x, y, t], {x, y, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
x:='x'; t:='t'; y:='y'; u:='u'; pde := diff(u(x, y, t), t, t) = (1/4)*(diff(u(x, y, t), x, x))+(1/4)*(diff(u(x, y, t), y, y)); bc := (D[1](u))(0, y, t) = 0, (D[1](u))(Pi, y, t) = 0, u(x, 0, t) = 0, u(x, Pi, t) = 0; ic:= u(x, y, 0) = x*y*(Pi-y),(D[3](u))(x, y, 0) = 0; sol:=pdsolve([pde,bc,ic],u(x,y,t)); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,bc,ic],u(x,y,t))),output='realtime')); sol:=subs(n1=m,sol);
\[ u \left ( x,y,t \right ) =\sum _{n=1}^{\infty }-2\,{\frac { \left ( -1+ \left ( -1 \right ) ^{n} \right ) \sin \left ( ny \right ) \cos \left ( 1/2\,nt \right ) }{{n}^{3}}}+\sum _{n=1}^{\infty } \left ( \sum _{m=1}^{\infty }-8\,{\frac { \left ( \left ( -1 \right ) ^{n+m}- \left ( -1 \right ) ^{n}- \left ( -1 \right ) ^{m}+1 \right ) \cos \left ( mx \right ) \sin \left ( ny \right ) \cos \left ( 1/2\,\sqrt {{m}^{2}+{n}^{2}}t \right ) }{{\pi }^{2}{m}^{2}{n}^{3}}} \right ) \]
____________________________________________________________________________________
Taken from Maple PDE help pages. This wave PDE inside square with damping present.
Membrane is free to move on the right edge and also on top edge. But fixed at left edge and bottom edge.
It has zero initial position, but given a non-zero initial velocity. Where \(0<x<1\) and \(0<y<1\) and \(t>0\).
Solve \[ \frac {\partial ^2 u}{\partial t^2} = \frac {1}{4} \left ( \frac {\partial ^2 u}{\partial x^2}+ \frac {\partial ^2 u}{\partial y^2} \right ) -\frac {1}{10} \frac {\partial u}{\partial t} \]
With boundary conditions
\begin {align*} u(0,y,t) &=0\\ \frac {\partial u}{\partial x}u(1,y,t) &= 0 \\ u(x,0,t) &=0 \\ \frac {\partial u}{\partial y}u(x,1,t) &= 0 \end {align*}
With initial conditions
\begin {align*} u(x,y,0) &=0 \\ \frac {\partial u}{\partial t}(x,y,0) &= x(1- \frac {1}{2} x) (1- \frac {1}{2} y) y \end {align*}
Mathematica ✗
ClearAll[u, t, y, x]; pde = D[u[x, y, t], {t, 2}] == (1*(D[u[x, y, t], {x, 2}] + D[u[x, y, t], {y, 2}]))/4 - (1*D[u[x, y, t], t])/10; ic = {u[x, y, 0] == 0, Derivative[0, 0, 1][u][x, y, 0] == x*(1 - (1/2)*x)*(1 - (1/2)*y)*y}; bc = {u[0, y, t] == 0, Derivative[1, 0, 0][u][1, y, t] == 0, u[x, 0, t] == 0, Derivative[0, 1, 0][u][x, 1, t] == 0}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, bc, ic}, u[x, y, t], {x, y, t}], 60*10]];
\[ \text {Failed} \]
Maple ✓
x:='x'; t:='t'; y:='y'; u:='u'; pde := diff(u(x, y, t), t$2) = 1/4*(diff(u(x, y, t), x$2)+diff(u(x, y, t), y$2))-(1/10)*(diff(u(x, y, t), t)); bc := u(0, y, t) = 0, (D[1](u))(1, y, t) = 0, u(x, 0, t) = 0, (D[2](u))(x, 1, t) = 0; ic:= u(x, y, 0) = 0, (D[3](u))(x, y, 0) = x*(1-(1/2)*x)*(1-(1/2)*y)*y; sol:=pdsolve([pde, ic,bc], u(x, y, t)); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic,bc], u(x, y, t))),output='realtime')); sol:=subs(n1=m,sol);
\[ u \left ( x,y,t \right ) =\sum _{m=0}^{\infty } \left ( \sum _{n=0}^{\infty }5120\,{\frac {{{\rm e}^{-t/20}}\sin \left ( 1/2\, \left ( 1+2\,m \right ) \pi \,y \right ) \sin \left ( 1/2\, \left ( 1+2\,n \right ) \pi \,x \right ) \sin \left ( 1/20\,t\sqrt {-1+ \left ( 100\,{m}^{2}+100\,{n}^{2}+100\,m+100\,n+50 \right ) {\pi }^{2}} \right ) }{\sqrt {-1+ \left ( 100\,{m}^{2}+100\,{n}^{2}+100\,m+100\,n+50 \right ) {\pi }^{2}}{\pi }^{6} \left ( 1+2\,m \right ) ^{3} \left ( 1+2\,n \right ) ^{3}}} \right ) \]
____________________________________________________________________________________
Taken from Mathematica helps pages on DSolve
Solve for \(u(x,y,t)\) with \(0<x<1\) and \(0<y<2\) and \(t>0\).
Solve \[ \frac {\partial ^2 u}{\partial t^2} = \frac {\partial ^2 u}{\partial x^2}+ \frac {\partial ^2 u}{\partial y^2} \]
With boundary conditions
\begin {align*} u(x,0,t) &=0 \\ u(0,y,t) &= 0 \\ u(1,y,t) &=0 \\ u(x,2,t) &= 0 \end {align*}
With initial conditions
\begin {align*} u(x,y,0) &=\frac {1}{10} (x-x^2)(2 y-y^2) \\ \frac {\partial u}{\partial t}(x,y,0) &= 0 \end {align*}
Mathematica ✓
ClearAll[u, t, y, x, n, m]; pde = D[u[x, y, t], {t, 2}] == Laplacian[u[x, y, t], {x, y}]; ic = {u[x, y, 0] == (1/10)*(x - x^2)*(2*y - y^2), Derivative[0, 0, 1][u][x, y, 0] == 0}; bc = {u[x, 0, t] == 0, u[0, y, t] == 0, u[1, y, t] == 0, u[x, 2, t] == 0}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic, bc}, u[x, y, t], {x, y, t}], 60*10]]; sol = sol /. {K[1] -> n, K[2] -> m}; sol = Assuming[Element[{n, m}, Integers], FullSimplify[sol]];
\[ \text {Bad latex generated} \]
Maple ✓
x:='x'; t:='t'; y:='y'; u:='u'; pde := diff(u(x, y, t), t$2) = diff(u(x, y, t), x$2)+diff(u(x, y, t), y$2); ic:=u(x,y,0)=(1/10)*(x-x^2)*(2*y-y^2),(D[3](u))(x,y,0)=0; bc:=u(x,0,t)=0,u(0,y,t)=0,u(1,y,t)=0,u(x,2,t)=0; sol:=pdsolve([pde, ic,bc], u(x, y, t)); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic,bc], u(x, y, t))),output='realtime')); sol:=subs(n1=m,sol);
\[ u \left ( x,y,t \right ) =\sum _{m=1}^{\infty } \left ( \sum _{n=1}^{\infty }-{\frac {32\,\sin \left ( n\pi \,x \right ) \sin \left ( 1/2\,m\pi \,y \right ) \cos \left ( 1/2\,\pi \,\sqrt {{m}^{2}+4\,{n}^{2}}t \right ) \left ( - \left ( -1 \right ) ^{m+n}+ \left ( -1 \right ) ^{m}+ \left ( -1 \right ) ^{n}-1 \right ) }{5\,{n}^{3}{\pi }^{6}{m}^{3}}} \right ) \]
____________________________________________________________________________________
Added Nov 27, 2018.
This is problem 8.5.5 part(a) from Richard Haberman applied partial differential equations 5th edition.
Solve the initial value problem for membrane with time-dependent forcing and fixed boundaries \(u=0\).
\[ \frac {\partial ^2 u(x,y,t)}{\partial t^2} = c^2 \nabla (u) + Q(x,y,t) \]
If the memberane is rectangle \((0<x<L,0<y<H)\).
With initial conditions
\begin {align*} u(x,y,0) &=f(x,y) \\ \frac {\partial u}{\partial t}(x,y,0) &= 0 \end {align*}
See my HW9, Math 322, UW Madison.
Mathematica ✗
ClearAll[u, t, y, x, n, m, L, H, Q, f]; pde = D[u[x, y, t], {t, 2}] == c^2*Laplacian[u[x, y, t], {x, y}] + Q[x, y, t]; ic = {u[x, y, 0] == f[x, y], Derivative[0, 0, 1][u][x, y, 0] == 0}; bc = {u[0, y, t] == 0, u[L, y, t] == 0, u[x, 0, t] == 0, u[x, H, t] == 0}; sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, ic, bc}, u[x, y, t], {x, y, t}, Assumptions -> {L > 0, H > 0, t > 0, c > 0}], 60*10]];
\[ \text {Failed} \]
Maple ✗
x:='x'; t:='t'; L:='L'; c:='c';u:='u';Q:='Q'; interface(showassumed=0); pde:=diff(u(x,y,t),t$2)=c^2*(diff(u(x,y,t),x$2)+diff(u(x,y,t),y$2))+Q(x,y,t); bc:=u(0,y,t)=0,u(L,y,t)=0,u(x,0,t)=0,u(x,H,t)=0; ic:=u(x,y,0)=f(x,y), eval( diff(u(x,y,t),t),t=0)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,bc,ic],u(x,y,t)) assuming L>0,H>0,c>0,t>0),output='realtime'));
\[ \text { sol=() } \]