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Heat Equation : Moving Source - Dirac Impulse Function

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Please see the attached file for the fully formatted problem.

Use superposition to solve:

with boundary conditions:

and initial condition

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Solution Summary

A heat equation with a moving source is investigated using the Dirac Impulse Function. The solution is detailed and well presented.

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Use superposition to solve:

with boundary conditions:

and initial condition

.

Solution:

Let's consider a more general problem of heat equation of form:
( 1)
with initial condition:
( 2)
The solution of this equation can be expressed as a superposition of 2 elemantary solutions:
( 3)
where
u1 = the general solution of homogenous associated equation of (1)
u2 = a particular solution of non-homogenous equation (1)
There are several methods to determine the solution of (1), classical and modern, using the theory of distributions.
I prefer to use the theory of distributions, so that I will try to sketch the steps to determine the complete solution.
We will use the following definitions and integral transforms:

1) The "fundamental solution" of (1), denoted by (E) is the solution of associated equation:
...

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