PDE with Time-Dependent Domain

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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)

Solution:

a) By using the variables change, we will have:
( 1)
As a result, our equation becomes:
( 2)
with boundary conditions:
( 3)
For an easier writing, we will not write anymore the "bar", by we know that we are working with transformed variables. At the end, we will take this into account.

There best method to determine the solution of (2) + (3) consists in using the Laplace integral transform.

We will use the following definitions and theorems:

1) h(x) = Heaviside function:
( 4)
2) (x) = Dirac distribution (sometimes called "Dirac impulse function") which satisfies the equation
( 5)
3) The "convolution product" between 2 functions (or distributions) is defined as ...