Page 82 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

72 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Substituting x = cos θ,

∂ 2 ∂ ∂u
2
r (ru) + (1 − x ) = 0 (4.146)
∂r 2 ∂x ∂x
We separate variables by assuming u = R(r)X(x). Substitute into the equation and divide by
RX and find

2

r [(1 − x )X ]
2 (4.147)
(rR) =− =±λ
R X
or
2
r(rR) ∓ λ R = 0

(4.148)
2
2
[(1 − x )X ] ± λ X = 0

The second of these is Legendre’s equation, and we have seen that it has bounded
2
solutions at r = 1 when λ = n(n + 1). The first equation is of the Cauchy–Euler type with
solution
n
R = C 1 r + C 2 r −n−1 (4.149)
Noting that the constant C 2 must be zero to obtain a bounded solution at r = 0, and using
superposition,

∞
n
u = K n r P n (x) (4.150)
n=0

and using the condition at fr = 1 and the orthogonality of the Legendre polynomial

π π

2 2K n
f (θ)P n (cos θ)dθ = K n P (cos θ)dθ = 2n + 1 (4.151)
n
θ=0 θ=0

4.4 ASSOCIATED LEGENDRE FUNCTIONS
Equation (1.15) in Chapter 1 can be put in the form
2
2
1 ∂u ∂ u 2 ∂u 1 ∂ ∂u 1 ∂ u
2
= + + (1 − µ ) + (4.152)
2
2
2
α ∂t ∂r 2 r ∂r r ∂µ ∂µ r (1 − µ ) ∂ 2
by substituting µ = cos θ.