Page 76 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

66 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
where the minus sign is chosen so that the function is bounded. The solution for T is exponential
and we recognize the equation for R as Bessel’s equation with ν = 0.

1
2
(ηR η ) η + λ R = 0 (4.111)
η
The solution is a linear combination of the two Bessel functions of order 0.

C 1 J 0 (λη) + C 2 Y 0 (λη) (4.112)

Since we have seen that Y 0 is unbounded as η approaches zero, C 2 must be zero. Furthermore,
the boundary condition at η = 1 requires that J 0 (λ) = 0, so that our eigenfunctions are J 0 (λη)
and the corresponding eigenvalues are the roots of J 0 (λ n ) = 0.

2
U n = K n e −λ τ J 0 (λ n η), n = 1, 2, 3, 4,... (4.113)
n
Summing (linear superposition)
∞
2
−λ τ
U = K n e n J 0 (λ n η) (4.114)
n=1
Using the initial condition,

∞

1 = K n J 0 (λ n η) (4.115)
n=1
Bessel functions are orthogonal with respect to weighting factor η sincetheyaresolutions to a
Sturm–Liouville system. Therefore when we multiply both sides of this equation by ηJ 0 (λ m η)dη
and integrate over (0, 1) all of the terms in the summation are zero except when m = n.Thus,
1 1

2
J 0 (λ n η)ηdη = K n J (λ n η)ηdη (4.116)
0
η=0 η=0
but

1
J (λ n )
2
2 1
ηJ (λ n η)dη =
0
2
η=0
1
1

ηJ 0 (λ n η)dη = J 1 (λ n ) (4.117)
λ n
η=0