DFT for partial differential equations

Question

I find it difficult to find the example of DFT (Discrete Fourier Transform) that solve partial differential equation. Would you mind to give me an example? From the equation that should solved until step by step solve it using DFT. Thank you so much.

Answer 1

The finite fourier transform is a technique possibly used when solving boundary value problems for linear PDE's.

Defining

$$S_n = \frac{2}{L}\int_0^L f(x)\sin{\frac{n \pi x}{L}} \, dx$$

to be the finite fourier sine-transform, it's inverse is given by

$$f(x) = \sum_{n=1}^{\infty} S_n \sin{\frac{n \pi x}{L}} $$

A problem appropriate for this method would be something like

$$u_{tt} = u_{xx} + \sin{\pi x}, u(0,t)=0, u(1,t)=0, u(x,0)=1,u_t(x,0)=0$$ on the domain $[0,1] \times [0,\infty)$.