# Derive Source Solution by Performing Integral Tranforms on a Heat Equation.

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Derive the source solution by performing integral transforms of the equation:

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Problem.

Derive the source solution by performing integral transforms of the equation:

Solution:

Let's consider the general homogenous Cauchy problem of heat equation:

( 1)

with initial condition:

( 2)

There best method to determine the solution of (1) + (2) consists in using the integral transforms (Laplace and Fourier).

We will use the following definitions and integral transforms:

1) (x) = Dirac distribution (sometimes called "Dirac impulse function") which satisfies the equation

( 3)

2) The "convolution product" between 2 functions (or distributions) is defined as

for 0 < t < ( 4)

or

for x R ( 5)

3) The Laplace transform:

( 6)

4) Laplace transform ...

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