Partial Differential Equations - Heat Equation

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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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