Clarke B. M. N., Fourier Theory.pdf

(326 KB) Pobierz
Math 335 Fourier Theory Notes
Macquarie University
Department of Mathematics
F o u r i e r T h e o r y
B . M . N . C l a r k e
374087887.006.png 374087887.007.png 374087887.008.png
Table of Contents
1.
Introduction ................................................................................................ .................................
1
2.
Linear differential operators ................................................................................................ ......
3
3.
Separation of variables ................................................................................................ ...............
5
4.
Fourier Series ................................................................................................ ..............................
9
5.
Bessel's inequality ................................................................................................ .....................
14
6.
Convergence results for Fourier series ....................................................................................
16
7.
Differentiation and Integration of Fourier Series ...................................................................
20
8.
Half-range Fourier series ................................................................................................ .........
23
9.
General Intervals ................................................................................................ .....................
25
10. Application to Laplace's equation ............................................................................................
27
11. Sturm-Liouville problems and orthogonal functions ..............................................................
33
12. Bessel Functions ................................................................................................ .......................
37
13. Fourier Transforms ................................................................................................ ..................
41
14. Inverse Fourier Transforms ................................................................................................ .....
47
15. Applications to Differential Equations ....................................................................................
50
16. Plancherel's and Parseval's Identities .....................................................................................
54
17. Band Limited Functions and Shannon's Sampling Theorem .................................................
56
18. Heisenberg’s Inequality ................................................................................................ ............
59
374087887.009.png
1. Introduction.
1
1. Introduction.
Fourier theory is a branch of mathematics first invented to solve certain problems in
partial differential equations. The most well-known of these equations are:
Laplace's equation,
x 2 +
y 2 = 0, for u ( x , y ) a function of two variables,
the wave equation,
2 u
t 2 – c 2
x 2 = 0, for u ( x , t ) a function of two variables,
the heat equation,
u
t –
k
x 2 = 0, for u ( x , t ) a function of two variables.
In the heat equation, x represents the position along the bar measured from some origin, t
represents time, u ( x , t ) the temperature at position x , time t . Fourier was initially
concerned with the heat equation. Incidentally, the same equation describes the
concentration of a dye diffusing in a liquid such as water. For this reason the equation is
sometimes called the diffusion equation.
In the wave equation, x , represents the position along an elastic string under tension,
measured from some origin, t represents time, u ( x , t ) the displacement of the string from
equilibrium at position x , time t .
In Laplace's equation, u ( x , y ) represents the steady temperature of a flat conducting plate
at the position ( x , y ) in the plane.
Since both the heat equation and the wave equation involve a single space variable x , we
sometime refer to them as the one dimensional heat equation and the one dimensional
wave equation respectively.
Laplace's equation involves two spatial variables and is therefore sometimes called the
two-dimensional laplace equation . Laplace's equation is connected to the theory of analytic
functions of a complex variable. If f ( z ) = u ( x , y ) + iv ( x , y ), the real and imaginary parts
u ( x , y ), v ( x , y ) satisfy the Cauchy-Riemann equations,
u
x =
v
y ,
u
y = –
v
x ,
Then
x 2 =
2 v
y = –
2 u
x
y 2
or
x 2 +
y 2 = 0.
2 u
Similarly,
x 2 +
y 2 = 0.
2 v
Heat conduction and wave propagation usually occur in 3 space dimensions and are
described by the following versions of Laplace's equation, the heat equation and the wave
equation;
x 2 +
y 2 +
2 u
z 2 = 0,
2 u
2 u
2 u
2 u
2 u
2 u
2 v
2 u
2 u
374087887.001.png 374087887.002.png
1. Introduction.
2
u
t –
k
æ
x 2 +
y 2 +
2 u
z 2 = 0,
ø ö
2 u
t 2 – c 2
æ
x 2 +
y 2 +
2 u
z 2 = 0.
ø ö
2 u
2 u
2 u
2 u
374087887.003.png
2. Linear differential operators.
3
2. Linear differential operators.
All of the above mentioned partial differential equations can be written in the form
L [ u ] = F
where
L [ u ]
º Ñ
2 u =
x 2 +
y 2 +
2 u
z 2 in Laplace's equation,
L [ u ]
º
t –
k
æ
x 2 +
2 u
y 2 +
2 u
2 u
z 2 =
ø ö
u
t –
k
Ñ
2 u in the heat equation,
and
L [ u ]
2 u
t 2 – c 2
æ
x 2 +
y 2 +
2 u
z 2 =
ø ö
2 u
t 2 – c 2
Ñ
2 u in the heat equation.
L [ u ] is in each case, a linear partial differential operator . Linearity means that for any
two functions u 1 , u 2 , and any two constants c 1 , c 2 ,
L [ c 1 u 1 + c 2 u 2 ] = c 1 L [ u 1 ] + c 2 L [ u 2 ].
In other words, L is linear if it preserves linear combinations of u 1 , u 2 . This definition
generalises to
L [ c 1 u 1 + .... + c n u n ] = c 1 L [ u 1 ] + ... + c n L [ u n ]
for any functions u 1 ,..., u n and constants c 1 ,..., c n .
Let u ( x 1 , x 2 ,..., x n ) be a function of n variables x = ( x 1 , x 2 ,..., x n ). Then the most general linear
partial differential operator is of the form
n
n
2 u
n
b i ( x )
u
L [ u ] =
i =1
j =1
a ij ( x )
x j +
i =1
x i + c ( x ) u
x i
where a ij ( x ), b i ( x ), c ( x ) are given coefficients.
The highest order partial derivative appearing is the order of the partial differential
operator. Henceforth we will consider only second order partial differential operators of the
form
n
n
2 u
n
b i ( x )
u
L [ u ] =
i =1
j =1
a ij ( x )
x j +
i =1
x i + c ( x ) u .
x i
The general linear second order partial differential equation is of the form
L [ u ] = F ( x )
where F ( x ) is a given function. When F ( x )
º
0, the equation L [ u ] = 0 is called homogeneous .
If F ( x )
¹
0, the equation L [ u ] = F ( x ) is called non-homogeneous .
Linearity of L is essential to the success of the Fourier method. There are usually
(infinitely) many solutions of a linear partial differential equation. The number of
2 u
2 u
u
2 u
2 u
º
374087887.004.png 374087887.005.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin