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Re: Laplace transform

• To: mathgroup at yoda.physics.unc.edu
• Subject: Re: Laplace transform
• From: Cetin Cetinkaya <cetin at acm0.me.uiuc.edu>
• Date: Thu, 26 May 1994 10:35:02 -0500

 
Subject: Numerical Laplace Transform

Some people expressed interest in  numerical Laplace Transform. In the following,
I will try to explain how I do it.

The standart way to do (Inverse) Laplace transformation is to use the FFT scheme.
The first step is to replace the Laplace variable s with j w (j=Sqrt[-1], w= angular
frequency). Then, the FFT scheme is used to transform the function in the frequency 
domain into real domain. This is rather simple process. 

Other issues involved with the FFT such as aliasing, leakage, truncation and windowing, 
and padding can be found in many books on the subject. Also, artificial dampling is an
inportant point.              

For the general use, one could find the following references useful: 

1- E.O. Brigham, The Fast Fourier Transforms and Its Applications, 1988, Prentice Hall.
(nice introduction)
2- R.E. Crandall, Mathematica for the Sciences, 1991,Addison-Wesley. (especialy, 
section 8.2 "Applications of the FFT" is very useful).
3- J.F. Doyle, Wave Propagation in Structures, 1989, Springer-Verlag. (many applications 
of the spectral methot to wave propagation problems).

I use the following two functions in my work:
Fo[list_]:=Sqrt[Length[list]]InverseFourier[list]//N;
InFo[list_]:= Fourier[list]/Sqrt[Length[list]]//N; 

I hope this will help.

Cetin Cetinkaya