# Partial Differential Equations - Heat Equation

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The causal solution f(x t) of the one-dimensional diffusion equation with source term f

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##### Solution Summary

The inhomogeneous heat equation (partial differential equation) in 1D can be solved by means of Green function

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The causal solution of the one-dimensional diffusion equation with source term f

(1)

is given by

(2)

where is the causal Green function satisfying

(3)

a) Change variables in (3) to , . Letting with the Heaviside step function , show (3) is solved if satisfies

(4)

Solve by Fourier transforming with respect to the position variable. Hence obtain an expression for

b) The temperature of a long uniform insulating metal rod satisfies

A heating element is used to create an initial temperature distribution Writing obtain an equation for , and combining (1), (2) and the answer to a) identify the temperature variation at the center of the rod, .

Solution:

a) After changing the variables, equation (3) becomes:

(5)

and putting

(6)

one yields:

(7)

where the following property of Dirac delta distribution was used:

(7')

The other partial derivative will be:

...

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