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# 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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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:
(8)
By replacing (7) and (8) in (5), we will have:
(9)
which can be written also as
(10)
This expression is equivalent to the following equations simultaneously satisfied:
(11)
(proof of statement (4))
One applies now the Fourier transform on the homogeneous equation (11) with respect to variable "z", using the formula:
(12)
and the property of derivative:
(13)
It follows that
(14)
and (15)
Applying the Fourier transform to the initial condition, we will get:
(16)
where the Fourier transform applied to Dirac delta distribution yields "1".
The equation (11) becomes a simple ordinary differential equation of 1'st order:
(17)
whose general solution is:
(18)
By applying the initial condition, it follows that C0 = 1, so that
(19)
The inverse of Fourier transform (12) is
(20)
and, if applied to (19), yields:

In the last integral, the variable change:
(21)
yields
(22)
Substituting in (6), the expression of Green function will be found:
(23)
or
(24)
b) It is required to solve the p.d. equation:
(25)
with the initial condition
(26)
One considers the function:
(27)
whose partial derivatives are:
(28)
(29)
It follows that

Since the last parenthesis is null (according to (25)), the equation to be solved is:
(30)
which is a inhomogeneous p.d. equation of general form
(31)
The solution of this kind of equation is expressed by formula:
(32)
where the Green function G is given by (24) and
(33)
Replacing with the appropriate expressions, the integral (32) becomes:
(34)
Using again the property (7') of Dirac distribution as well as the property
(35)
the integral (34) becomes:

(36)
Comparing to (27), one deduces that
(37)
The temperature in the center of the rod will be given by:
(38)
Using a new variable change, the Poisson integral will be again obtained:
(39)

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