Module for Convolution Method for Laplace Transforms

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Background.  (Convolution Theorem)

    Let    denote the Laplace transforms of  ,  respectively.  Then the product is the Laplace transform of the convolution of  ,  and is denoted by  ,  and has the integral representation
    
    ,
    or
    .

Example 1.  Use convolution to find the inverse Laplace transform of  .  

Example 2.  Use the convolution theorem to solve the integral equation  

    .  

More Background.

The Laplace transform of the Dirac delta function    is  .  
This can be illustrated with Mathematica.

Example 3.  Solve the initial value problem  

  with   .  

Example 4.   Use convolution to solve the initial value problem  

  with  .  

Plot the solution over the interval  .