Fourier Analysis for numeric DE

Hi,

I have to do some analysis of finite difference methods using Fourier
Analysis. This is problematic because I have not studied Fourier
Analysis--the only thing I've done is approximate functions with a Fourier
series as part of a lab for a lower division ODE course.

In one of my textbooks, they write: "We can now easily show that a similar
Fourier mode is an exact solution of the difference equations. Suppose we
substitute:

U_n,j = (lambda)^n * exp( i*k*(j*dx) )

into the difference equation

U_n+1, j = U_n,j + mu*(U_n,j+1 - 2U_n,j + u_n,j-1)

where mu = dt / (dx)^2, and putting

U_n+1 = lambda*U_n,j and similarly for the other terms.

Let me stop there for now. In particular, I don't get where

U_n,j = (lambda)^n * exp( i*k*(j*dx) )

comes from. What is lambda? Why do this substitution?

>From what I gather, we are trying to write the finite difference iteration
in a Fourier form, and then study the method in this form to find out where
it is stable.


.