Can the Forced Wave Equation Be Solved Numerically?

Tohiko
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Hi,

I want to solve the following wave equation:
u_{tt} - c^2 u_{xx} = f(x,t)u

What is the best way to do it? I don't think I can use Duhamel's principle since I have a u in the forcing.
Doing a change of variables of the form
w=x+ct, v=x-ct
Seems to make things worse.

Any ideas?
Thank you
 
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I doubt there is a closed form solution for general f(x,t).

Even

\frac{d^2y}{dx^2} + [a + 2q\cos(2x)]y = 0

doesn't have a closed form solution. (This is known as the Mathieu equation).
 
Yes, I've worked with Mathieu's Equations before

And actually I think I might be working with some form of them since in one of the
equations that I want to solve I have
f(x,t) = 1- a \cos (b t)
For some constants a,b

Back to the general forced wave equation: can it be solved numerically? If so, can you give me some pointers on how to do that?
 
Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
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