Page 81 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 71
The second solution of Legendre’s equation can be found by the method of variation of
parameters. The result is

dζ

Q n (x) = P n (x) (4.141)
2
2
P (ζ)(1 − ζ )
n
It can be shown that this generally takes on a logarithmic form involving ln [(x + 1)/(x − 1)]
which goes to infinity at x = 1. In fact it can be shown that the first two of these
functions are

1 1 + x x 1 + x
Q 0 = ln and Q 1 = ln − 1 (4.142)
2 1 − x 2 1 − x

Thus the complete solution of the Legendre equation is

u = AP n (x) + BQ n (x) (4.143)

where P n (x)and Q n (x) are Legendre polynomials of the first and second kind. If we require
the solution to be finite at x = 1, B must be zero.
Referring back to Eqs. (3.46) through (3.53) in Chapter 3, we note that the eigenvalues
λ = n(n + 1) and the eigenfunctions are P n (x)and Q n (x). We further note from (3.46) and
(3.47) that the weight function is one and that the orthogonality condition is

1
2

P n (x)P m (x)dx = δ mn (4.144)
2n + 1
−1
where δ mn is Kronecker’s delta, 1 when n = m and 0 otherwise.

Example 4.13. Steady heat conduction in a sphere

Consider heat transfer in a solid sphere whose surface temperature is a function of θ, the angle
measured downward from the z-axis (see Fig. 1.3 Chapter 1). The problem is steady and there
is no heat source.

∂ 2 1 ∂ ∂u
r (ru) + sin θ = 0
∂r 2 sin θ ∂θ ∂θ
u(r = 1) = f (θ) (4.145)

u is bounded