PDE and Laplace Transform Examples

This is an evaluated Mathematica notebook. Since the notebook is intended to be interactive, it may be helpful to also view the unevaluated version


Copyright 1996
Gary S. Stoudt
Mathematics Department
Indiana University of PA
Indiana, PA 15705
GSSTOUDT@grove.iup.edu

This notebook has some examples of using Laplace transforms to solve certain PDE's. There is an animation of a solution-this accounts for the size of the file.

In[1]:=

  <<Calculus`LaplaceTransform`

Throughout this notebook, we use ut for the partial of u(x,t) with respect to t and uxx for the second partial of u(x,t) with respect to x. We also use a capital letter to denote the transformed function, and s to denote the transformed variable.

It is often easy to apply the Laplace transform to the PDE. What is important to remember is that we are transforming in the time variable t. This means that, for example,
LaplaceTransform[ut]=sU(x,s)-u(x,0)
LaplaceTransform[uxx]=Uxx.

Example 1

Example 2

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