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Fourier Sine and Cosine Transformations: Module 8: The Fourier Transform Methdos For Pdes

The document discusses the Fourier sine and cosine transforms and their properties. These transforms are useful for problems over semi-infinite intervals where the function or its derivative are prescribed on the boundary. The Fourier transform of an even function is called the Fourier cosine transform, while the Fourier transform of an odd function is called the Fourier sine transform. Basic properties of the Fourier cosine and sine transforms are presented, including linearity and transformations of derivatives. Practice problems are provided to test understanding.

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MohammedHassanGomaa
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56 views3 pages

Fourier Sine and Cosine Transformations: Module 8: The Fourier Transform Methdos For Pdes

The document discusses the Fourier sine and cosine transforms and their properties. These transforms are useful for problems over semi-infinite intervals where the function or its derivative are prescribed on the boundary. The Fourier transform of an even function is called the Fourier cosine transform, while the Fourier transform of an odd function is called the Fourier sine transform. Basic properties of the Fourier cosine and sine transforms are presented, including linearity and transformations of derivatives. Practice problems are provided to test understanding.

Uploaded by

MohammedHassanGomaa
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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MODULE 8: THE FOURIER TRANSFORM METHDOS FOR PDES

Lecture 2

Fourier Sine and Cosine Transformations

In this lecture we shall discuss the Fourier sine and cosine transforms and their properties.
These transforms are appropriate for problems over semi-innite intervals in a spatial
variable in which the function or its derivative are prescribed on the boundary.
If a function is even or odd function then f can be represented by a Fourier integral
which takes a simpler form than in the case of an arbitrary function.
If f (x) is an even function, then B() = 0 in (3), and

A() = 2
f (t) cos tdt.
0

Hence, the Fourier integral reduces to the simpler form

1
f (x) =
A() cos(x)d.
0
Similarly, if f (x) is odd, then A() = 0 in (3), and

f (t) sin tdt.
B() = 2
0

Thus, (3) becomes

1
f (x) =
B() sin(x)d.
0
These Fourier integrals motivates to dene the Fourier cosine transform (FCT) and Fourier
sine transform (FST). The FT of an even function f is called FCT of f . The FT of an
odd function f is called the FST of f .
DEFINITION 1. (Fourier Cosine Transform) The FCT of a function f : [0, ) R
is dened as
Fc (f ) = fc () = Fc () =


2
f (x) cos(x)dx (0 < ).
0

(1)

DEFINITION 2. (Inverse Fourier Cosine Transform ) The Inverse FCT (IFCT) of a


function fc () (0 < ) is dened as

2
1
Fc [fc ] = fc (x) =
fc () cos(x)d (0 x < ).
(2)
0
DEFINITION 3. (Fourier Sine Transform) The FST of a function f : [0, ) R is
dened as

2
Fs (f ) = fs () = Fs () =
f (x) sin(x)dx (0 < ).
0

(3)

MODULE 8: THE FOURIER TRANSFORM METHDOS FOR PDES

DEFINITION 4. (Inverse Fourier Sine Transform) The Inverse FST (IFST) of a


function fs () (0 < ) is dened as
Fs1 (f )


2
= fs (x) = Fs () =
fs () sin(x)d (0 x < ).
0

(4)

Basic Properties of Fourier Cosine and Sine Transforms:


Linearity:
Fc [(af + bg)] = aFc [f ] + bFc [g].
Fs [(af + bg)] = aFs [f ] + bFs [g].
Let f be a function dened for x 0 and f (x) 0 as x . Then

2
sin(x)f (x)dx
Fs [f (x)] =
0
x=



2
2

=
sin(x)f (x)
cos(x)f (x)dx

0
x=0

= Fc [f ].
If we assume that f (x), f (x) then
x=



2
2
2

sin(x)f (x)dx =
sin(x)f (x)
cos(x)f (x)dx

0
x=0
x=

x=

2
2

sin(x)f (x)
cos(x)f (x)
=
+

x=0
x=0

2
sin(x)f (x)dx
2
0

2
=
f (0) 2 Fs [f ]

Thus, we have
Fs [f (x)] = Fc [f ].

Fs [f (x)] = Fs [f ] +
2

2
f (0).

A similar result is true for the Fourier cosine function.

Fc [f (x)] = Fs [f ]
f (0)

2
Fc [f (x)] = Fc [f ]
f (0).

MODULE 8: THE FOURIER TRANSFORM METHDOS FOR PDES

Note: Observe that the FST of a rst derivative of a function is given in terms of
the FCT of the function itself. However, the FST of a second derivative is given in
terms
of the sine transform of the function. There is an additional boundary term
2 f (0).
Transformation of partial derivatives:
(i) Let u = u(x, t) be a function dened for x 0 and t 0. If u(x, t) 0 as
x , and Fs [u](, t) = u
s (, t), then
Fs [ux ](, t) = Fc [u](, t).

2
Fc [ux ](, t) = Fs [u](, t)
u(0, t).

If, in addition, ux (x, t) 0 as x , then

2
u(0, t).
Fs [uxx ](, t) = 2 Fs [u](, t) +

2
2
Fc [uxx ](, t) = Fc [u](, t)
ux (0, t).

(ii) If we transform the partial derivative ut (x, t) (and if the variable of integration
in the transformation is x), then the transformation is given by
d
Fs [ut ](, t) = {Fs [u]}(, t).
dt
d
Fc [ut ](, t) = {Fc [u]}(, t).
dt
Thus, time dierentiation commutes with both the Fourier cosine and sine transformations.

Practice Problems
1. Find the FST and FCT of the function
{
1, 0 x 2,
f (x) =
0, x > 2.
2. If u = u(x, t) and u(x, t) 0 as x , then
(A) Fs ux (, t) = Fc [u](, t)
(B) Fc ux (, t) = 2 u(0, t) + Fs [u](, t)
3. If u(x, t) and ux (x, t) 0 as x , then

(A) Fs [uxx ](, t) = 2 Fs [u](, t) + 2 u(0, t)

(B) Fc [uxx ](, t) = 2 Fc [u](, t) 2 ux (0, t)

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