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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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