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

Please see the attached file for the fully formatted problems.

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

© BrainMass Inc. brainmass.com December 24, 2021, 5:10 pm ad1c9bdddfhttps://brainmass.com/math/integrals/derive-source-solution-performing-integral-transforms-heat-equation-33391

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

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 ...

#### Solution Summary

The Source Solution is derived by Performing Integral Tranforms on a Heat Equation. The solution is detailed and well presented.