BESSEL’S EQUATION AND BESSEL FUNCTIONS :

BESSEL’S EQUATION AND BESSEL FUNCTIONS:

The differential equation

(1) where n is a parameter, is called Bessel’s Equation of Order n.Any solution of Bessel’s Equation of Order n is called a Bessel Function of Order n.

0222

22 ynx

dxdyx

dxydx

Bessel’s Equation and Bessel’s Functions occur in connection with many problems of physics and engineering, and there is an extensive literature dealing with the theory and application of this equation and its solutions.

If n=0 Equation (1) is equivalent to the equation

(2)

which is called Bessel’s Equation of Order Zero.

02

2

xydxdy

dxydx

where A and B are arbitrary constants, and is called the Bessel Function of the First Kind of Order Zero. is called the Bessel Function of the Second Kind of Order Zero.

)()( 00 xYBxJAy

The general Solution of Equation (2) is given by :

0J

0Y

The functions J0 and Y0 have been studied extensively and tabulated. Many of the interesting properties of these functions are indicated by their graphs.

where A and B are arbitrary constants, andis called the

Bessel Function of the First Kind of Order n.

)()( xYBxJAy nn

The general Solution of Equation (1) is given by :

nJ

Bessel Functions of the first kind of order n

is called the Gamma Function

1!0,.....2,1,0;!)1()()1(

wherenifnnnnn

0)1()(

nfornnn

0)(0

1 nfordtetn tn

Bessel Functions of the first kind of order n

For n=0,1 we have

is called the Bessel Function of the Second Kind of Order n.

is Euler’s Constant and is defined by

nY

5772.0ln1...31

211lim

n

nn

For n=0

General Solution of Bessel Differential Equation

Generating Function for Jn(x)

Recurrence Formulas forBessel Functions

Bessel Functions of Order Equal to Half and Odd Integer

In this case the functions are expressible in terms of sines and cosines.

For further results use the recurrence formula.

Bessels Modified Differential Equations

Solutions of this equation are called modified Bessel functions of order n.

Modified Bessels Functions of the First Kind of Order n



Modified Bessels Functions of the Second Kind of Order n



General Solution of Bessel’s Modified Equation

Generating Function for In(x)

Recurrence Formulas for Modified Bessel Functions

Recurrence Formulas for Modified Bessel Functions

Modified Bessel Functions of Order Equal to Half and Odd Integer

In this case the functions are expressible in terms of hyperbolic sines and cosines.

For further results use the recurrence formula. Results for are obtained from



Graphs of Bessel Functions

0)0(1)0(

1

0

JJ

)0()0(

1

0

YY

1)0(0)0(1

oII

)0()0(

1

0

KK

Indefinite Integrals Involving Bessel Functions




Definite Integrals Involving Bessel Functions

Definite Integrals Involving Bessel Functions

Many differential equations occur in practice that are not of the standars form but whose solutions can be written in terms of Bessel functions.

A General Differential Equation Having Bessel Functions as Solutions



has the solution

Where Z stands for J and Y or any linear combination of them, and a, b, c, p are constants.

0212

22221

y

xcpabcxy

xay c

)( cp

a bxZxy

ExampleSolve y’’+9xy=0Solution:

0

;1)1(2;9)(

;021

222

2

cpa

cbc

a

From these equations we find

Then the solution of the equation is

3/1/;2;2/3;2/1

capbca

)2( 2/33/1

2/1 xZxy

This means that the general solution of the equation is

where A and B are constants

)2()2([ 2/33/1

2/33/1

2/1 xBYxAJxy



If

has the solution

0)1(

)2(222

2

yxbxpabdxc

ybxaxyxppq

p

zeronotisqorpanddandca 4)1( 2

)()( qqx xBZxAZexyp

qca

qd

pb

a

24)1(

;

;

;21

2

KId

IId

YJd

JJd

ZZd

00

00

00

00



If

has the solution

02

ybxax

dxdyx

dxd rsr

224)1( 2 borrsandbr

)()(

xBZxAZxy

srbr

sra

sr

r

24)1(

;22

;2

2

;21

2

KIa

IIa

YJa

JJa

ZZa

00

00

00

00

Problem

A pipe of radius R0 has a circular fin of radius R1 and thickness 2B on it (as shown in the figure below). The outside wall temperature of the pipe is Tw and the ambient air temperature is Ta. Neglect the heat loss from the edge of the fin (of thickness 2B). Assume heat is transferred to the ambient air by surface convection with a constant heat transfer coefficient h.

• a) Starting with a shell thermal energy balance, derive the differential equation that describes the radial temperature distribution in the fin.

• b) Obtain the radial temperature distribution in the circular fin.

• c) Develop an expression for the total heat loss from the fin.

Solution

From a thermal energy balance over a thin cylindrical ring of width Dr in the circular fin, we get Rate of Heat In - Out + Generation =

Accumulation The accumulation term (at steady-state) and the generation term will be zero. So,

where h is the (constant) heat transfer coefficient for surface convection to the ambient air and qr is the heat flux for conduction in the radial direction. Dividing by 4p B Dr and taking the limit as Dr tends to zero,

0)()2(2)22()22( DD arrrrr TThrrBqrBqr ppp

)()()(

lim0 a

rrrrr

rTTr

Bh

rrqrq

D

D

D

If the thermal conductivity k of the fin material is considered constant, on substituting Fourier’s law we get

Let the dimensionless excess temperature be denoted by q = (T - Ta)/(Tw - Ta). Then,

)()( ar TTrBhrq

drd

)()( aTTrkBh

drdTr

drd

0)( qq rkBh

drdr

drd

)/(;0;1;1

02

kBhabsr

ybxaxdxdyx

dxd rsr

211

2

)0(40114)1( 22

rsand

br

0

;1;0

a

)()( 000 xaBZxaAZx q

)()( 00 xaBKxaAI q

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BESSEL’S EQUATION AND BESSEL FUNCTIONS :