From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Fourier Transforms
Date: 1 Sep 1999 14:51:42 -0400
Newsgroups: sci.math

In article <37ccc84c_2@news1.one.net>,
Sandy Barnabas <sandy@webdice.net> wrote:
:Hi,
:
:Recently, I came across references to Fourier Transforms and Inverse
:Fourier Transforms....I know that they are somehow related to integral
:calculus, but not much more than that.  Can anyone point me to a resource
:that will explain these to me using nothing more complex than
:differential equations?
:

It is the effect of Fourier Transform on derivatives that makes it a tool
for (linear) differential equation:

If the Fourier Transform of a function f is denoted F, then the Fourier
transform of f'(t) is (-i*s)*F(s).

So, the transform converts differential equations to algebraic equations,
for example (with similar notation about g and G)

  f''(t) + 2 * f'(t) + 4 * f(t) = g(t)

becomes

  ((-i*s)^2 + 2 * (-i*s) + 4) * F(s) = G(s)

that is

  ( -s^2 - 2*i*s + 4) * F(s) = G(s)

You can now divide by the quadratic polynomial to get F(s) ; to finish,
you take the inverse Fourier Transform of G(s)/(the quadratic polynomial).

It is more fun if you transform partial differential equations, with
respect to one or more variables.

Kreyszig's textbook of mathematics for engineers will give you plenty of
applications, if I remember right.

Good luck, ZVK(Slavek).