Post on 25-Feb-2016
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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 First Kind of Order n
Modified Bessels Functions of the First Kind of Order n
Modified Bessels Functions of the Second Kind of Order n
Modified Bessels Functions of the Second 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
Modified Bessel Functions of Order Equal to Half and Odd Integer
Modified Bessel Functions of Order Equal to Half and Odd Integer
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
Indefinite Integrals Involving Bessel Functions
Indefinite Integrals Involving Bessel Functions
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
A General Differential Equation Having Bessel Functions as Solutions
The differential equation
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
A General Differential Equation Having Bessel Functions as Solutions
The differential equation
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
A General Differential Equation Having Bessel Functions as Solutions
The differential equation
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