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PDE with Time-Dependent Domain

Please see the attached file for the fully formatted problems.

Consider the diffusion equation:

on the time-dependent domain

where a is a constant. We wish to solve the initial and boundary value problem having

for and a prescribed . Thus, u is prescribed as a function of time on the left boundary that moves at a constant speed a.

a) Introduce the transformation of variables

and solve the resulting problem by Laplace transforms.

b) Calculate the appropriate Green's function for the problem in x, t variables and
rederive the solution using this.

This problem is taken from Partial Differential Equations: Analytical Solution Techniques, by J. Kevorkian. (Prob. 1.4.7)

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A PDE with Time-Dependent Domain is investigated using convolution and the the Dirac distribution. The solution is detailed and well presented. The response received a rating of "5" from the student who posted the question.

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