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Re: 2+ D wave equation


On 17 Feb 05 23:46:19 -0500 (EST), mohammad kazim wrote:
>I'm facing the same problem someone else posted (pasted below)about 6
>years ago. Unfortunately, nobody responded back then.
>
>I'm trying to numerically obtain the answer for a vibrating
>rectangular membrane. Therefore, I need to describe a 2 spatial
>dimensional wave equation, and I'm trying to use NDSolve.
>
>Thanks!
>
>On 5 Mar 1999 02:24:26 -0500, Robert Walgate wrote:
>>MathGroup
>>
>> A question: I need to find numerical solutions of the wave equation
>in 2
>>or more spatial dimensions, with various boundary conditions.
NDSolve
>>limits the problem to one spatial dimension. Has anyone created a
>package
>>to go to higher dimensions?
>>
>>(Dr) Robert Walgate
>>Open Solutions


NDSolve isn't limited to 1D equations, at least in version 5.1.
For example, this solves a 2D wave equation:
First[NDSolve[{
      D[u[x, y, t], {t, 2}] ==
        D[u[x, y, t], {x, 2}] +
          D[u[x, y, t], {y, 2}], u[x, y, 0] == 0,
      Derivative[0, 0, 1][u][x, y, 0] == -0.5*x(1 - x)*y(1 - y),
      u[0, y, t] == 0, u[1, y, t] == 0, u[x, 0, t] == 0,
      u[x, 1, t] == 0}, u, {x, 0, 1}, {y, 0, 1}, {t, 0, 10}]]
NDSolve is though limited to finite differences (on rect. regions),
but the implementation of FD is very sophisticated.

Regards
Jaroslav Hajek


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