Page 561 - Modelling in Transport Phenomena A Conceptual Approach

B.2. SECOIVD-ORDER LINEAR DLFFER.ENT1A.L EQUATIONS 541

Comparison of Eq. (8) with Eq. (B.2-16) gives p = 1; a = -1; b = - a; j = 1;
and k = - 1. Since k = p - 2, then Eq- (8) is Bessel’s equation.

B.2.3.1 Solution of Bessel’s equation

If an ordinary differential equation is reducible to the Bessel’s equation, then the
constants a, /3, and n are defined by
2-p+j
a= (B.2-17)
2
1-P (B .2- 18)
’=2-p+j
J( 1 - p)2 - 4b
n= (B.2-19)
2-p+j
The solution depends on whether the term a is positive or negative.

Case (i) a > 0

In this case the solution is given by

y = zap [CIJn(Rz*) + C2J-n(Rza)] n # integer (B.2-20)

y = zap [CIJn(nza) + C2Yn(0za)] n = integer (B.2-21)

where Cl and C2 are constants, and R is defined by

n=- JTi (B.2-22)
a

The term Jn(z) is known as the Bessel function of the first kind of order n and is
given by
(B.2-23)

J-n(z) is obtained by simply replacing n in Eq. (B.2-23) with - n. When n is not
an integer, the functions Jn(z) and J-,(z) are linearly independent solutions of
Bessel’s equation as given by Eq. (B.2-20). When n is an integer, however, these
two functions are no longer linearly independent. In this case, the solution is given
by Eq. (B.2-21) in which Yn(x) is known as Weber’s Bessel function of the second
kind of order n and is given by
(c0sn7r)Jn(z) - J-&)
Yn(4 = sin nr (B .2-24)