Can the Forced Wave Equation Be Solved Numerically?

[Tohiko]

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

 

[Mute]

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).

 

[Tohiko]

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?