12.6 The Fourier-Bessel Series Math 241 -Rimmer 2 …rimmer/math241/ch12sc6frbess.pdf · 3 [ ] 2 2...

1

12.6 The Fourier-Bessel SeriesMath 241 - Rimmer

( )2 2 2 2 0

parametric Bessel equation of order

x y xy x yα ν

ν

′′ ′+ + − =

( )

( ) ( )1 2

has general solution on 0, of

y c J x c Y xν να α

∞

= +

very important in the study of boundary-

value problems involving partial

differential equations expressed in

cylindrical coordinates

( ) is called a of order .J xν νBessel function of the first kind

( )( )( )

2

0

1

! 1 2

n n

n

xJ x

n n

ν

νν

+∞

=

− =

Γ + + ∑

( )( ) ( ) ( )

( )

For non-integer values of

cos

sin

J x J xY x

ν ν

ν

ν

νπ

νπ

−−=

( ) is called a of order .Y xν νBessel function of the second kind

( ) ( )

For integer values (say )

limnn

n

Y x Y xνν →

=


( ) : of order .nJ x nBessel function of the first kind

Let 0,1, 2,n nν = = …

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

2


http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html

( ) : of order .nY x nBessel function of the second kind

Let 0,1, 2,n nν = = …

( )2 2 2 2

The parametric Bessel differential equation becomes

0x y xy x n yα′′ ′+ + − =


Let 0,1, 2,n nν = = …

( )

( ) ( )1 2

with general solution on 0, of

n ny c J x c Y xα α

∞

= +

[ ]2

2

The self-adjoint form of the differential equation i :

0

s

d nxy x y

dx xα

′ + − =

( ) ( )

( )

2

2

we can identify

, and

,

,n

r x x q xx

p x x λ α

= = −

= =

This is called a singular Sturm-Liouville problem

when we add the following conditions:

( )

( ) ( )2 2

0 and instead of 2 boundary

conditions we only have

0

r a

A y b B y b

=

′+ =

For orthogonality, we need the

solutions to be bounded at .x a=

( )

( ) ( )2 2

and we will only need

0

0 0

A y b

r

B y b

=

′+ =

( )

( )

but 0 is unbounded

for all , so for orthogonality

we can only use

n

n

Y

n

J xα

3

[ ]2

2

The self-adjoint form of the differential equation i :

0

s

d nxy x y

dx xα

′ + − =


( ) ( )

( )

2

2

we can identify

, and

,

,n

r x x q xx

p x x λ α

= = −

= =

( )

( )1

with general solution

on 0, of

ny c J xα

∞

=

( ) ( ) ( ){ } ( )

( ) [ ]

2

1 2

this gives a set of orthogonal functions

, , , 1, 2,

that are orthogonal with respect to the weight function

on the interval 0,

n n n i i iJ x J x J x i

p x x b

α α α λ α= =

=

… … …

( ) ( )0

0,

b

n i n j i jxJ x J x dxα α λ λ= ≠∫

( ) ( )2 2

but this is all contingent upon the

being defined by a boundary condition

at of the type

0

i

n n

x b

A J b B J b

α

α α

=

′+ =

( ) ( ) ( ) ( )

by the chain rule

n n n

dJ x J x x J x

dxα α α α α′ ′ ′= =

so the condition becomes:

( ) ( )2 20

n nA J b B J bα α α′+ =


( )

So now for 0,1, 2, ,we have the Bessel functions of order

that will serve as our set of orthogonal functions used in the

eigenfunction expansion of :

n n

f x

= …

( ) ( ) ( ){ }

( ) [ ]

2 1 2 2 2

Let 2 for instance

, , , is a set of orthogonal

that are orthogonal with respect to the weight function

on the interval 0,

i

n

J x J x J x

p x x b

α α α

=

=

eigenfunctions… …

( )We are interested in taking a function and expanding

it using Fourier eigenfunction expansion.

f x

2with corresponding 1,2,i i iλ α= =eigenvalues …

( ) ( )1

i n i

i

f x c J xα∞

=

=∑

( ) ( ){ }The expansion of with Bessel functions 1,2,

is called a .

n if x J x iα =

−Fourier Bessel series

…

( ) ( )

( )0

2where

b

n i

i

n i

xJ x f x dx

cJ x

α

α=∫

4


In order to find the coefficients , we need 3 properties

of the Bessel function:

ic

J

( ) ( )12. n n

n n

dx J x x J x

dx−

=

( ) ( ) ( )1. 1n

n nJ x J x− = −

( ) ( )13. n n

n n

dx J x x J x

dx

− −

+ = −

Three different versions of the boundary condition at

lead to three different types of solutions

x b=

( )1. 0nJ bα =

( ) ( )2. 0n n

hJ b bJ bα α α′+ =

( )03. 0J bα′ =

( )2

we'll have 3 different results for n iJ xα


( ) ( )1

i n i

i

f x c J xα∞

=

=∑

( )( ) ( )22

01

2b

i n i

n i

c xJ x f x dxb J b

αα+

=

∫

( )

when the defined by

the boundary condition 0n

i

J bα

α

=

( )

( )

2

2

:

#8 , 0 1

0

f x x x

J α

= < <

=

example( )

( )( )

1

3

22

03

21, 2,

2i i

i

c x J x

n

dxJ

b f x x

αα

=

= = =

∫

5


( ) ( )1

i n i

i

f x c J xα∞

=

=∑

( ) ( )( ) ( )

2

22 2 2 20

2b

ii n i

i n i

c xJ x f x dxb n h J b

αα

α α=

− + ∫

( ) ( )

when the defined by

the boundary condition

0

i

n nhJ b bJ bα α

α

α′+ =

( )

( ) ( )0 0

#6 1, 0 2

:

2 2 0

f x x

J Jα α α

= < <

′+ =

example


( ) ( )1 0

2

i i

i

f x c c J xα∞

=

= +∑

( )( ) ( )022

00

2b

i i

i

c xJ x f x dxb J b

αα

=

∫

( )0

when the defined by

the boundary condition 0

i

J b

α

α′ =

( )1 2

0

2b

c xf x dxb

= ∫

( )

( )0

#4 1,

:

0 2

2 0

f x x

J α

= < <

′ =

example

6


( )

( )

2

2

:

#8 , 0 1

0

f x x x

J α

= < <

=

example

( )( )

1

3

22

03

2i i

i

c x J x dxJ

αα

=

∫ ilet t xα=

1i

i

dt dx dx dtαα

= ⇒ =3

3

3

i i

t tx x

α α= ⇒ =( )

( )3

22 3

03

2 i

i

i ii

t dtc J t

J

α

α αα=

∫

( )( )3

22403

2 i

i

i i

c t J t dtJ

α

α α=

∫

( ) ( )1

n n

n n

dx J x x J x

dx−

=

( ) ( )3 3

3 2

dt J t t J t

dt ⇒ =

( )( )3

32403

2 i

i

i i

dc t J t dt

dtJ

α

α α =

∫

0 0

1 i

x t

x t α

= ⇒ =

= ⇒ =

( )( )3

32 04

3

2 i

i

i i

c t J tJ

α

α α =

( )

( )

3

3

24

3

2 i i

i i

J

J

α α

α α

=

( )3

2i

i i

cJα α

= ( ) ( ) ( )3

12

1

2i i

iJi

f x J xα α

α∞

=

= ∑

12.6 The Fourier-Bessel Series Math 241 -Rimmer 2 …rimmer/math241/ch12sc6frbess.pdf · 3 [ ] 2 2...

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