Transcript MATH2965

Solving PDEs using Fourier transforms
Fourier transforms can be used to solve PDEs if one of the
variables has an infinite range.
In this case a PDE can be transformed into an ODE.
Example – Solve
subject to
Define:
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Solving PDEs using Fourier transforms
Next, take the Fourier transform of BOTH SIDES of the PDE:
Now,
Therefore
or
an ODE.
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Solving PDEs using Fourier transforms
This a simple ODE (1st order, linear, constant coeff.) of the form
Therefore
(1)
Now take the FT of the given initial condition:
But from (1)
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Solving PDEs using Fourier transforms
Substituting into (1) gives
Finally, taking the inverse transform and using the table,
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MATH2065 Introduction to PDEs
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Heat equation using Fourier transforms
Use Fourier transforms to solve the 1-D Heat equation
with initial condition
There are implicit physical conditions such as
Take the FT with respect to x.
Define
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Heat equation using Fourier transforms
Then
Substitute into the PDE
Simple ODE:
Solving the ODE
(*)
Inverting
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Heat equation using Fourier transforms
Letting
Then, the initial condition
from which we can calculate
Ie,
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MATH2065 Introduction to PDEs
gives
by inversion.
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Heat equation using Fourier transforms
Summarising, the solution of the heat equation in the infinite
domain
with IC
is given by
where
We can now rewrite this as a convolution, but alternatively we can
use convolution in (*) from the outset:
Recall that
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Heat equation using Fourier transforms
Hence,
But
and (from the inverse of a Gaussian):
Therefore
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Heat equation using Fourier transforms
In the special case where
we get
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Tutorial problem
Use Fourier transforms to show that the solution of
with initial condition
is:
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Inverse Fourier transform of a Gaussian
To find the solution of the heat equation and other problems, we
will need to find the inverse Fourier transform of the function
(cf. Page 131 of Notes)
This is the well-known bell-shaped curve known as a Gaussian.
By definition, the inverse transform is given by
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Inverse Fourier transform of a Gaussian
To evaluate this integral we use the following “trick”:
First, differentiate with respect to x
Next, integrate by parts:
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Inverse Fourier transform of a Gaussian
The first term vanishes as
Therefore
Ie,
This is a simple separable first order ODE for g(x).
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Inverse Fourier transform of a Gaussian
But
and it can be shown that
Therefore, the final expression for the inverse transform of the
Gaussian is:
which itself is another Gaussian.
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