Page 49 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

ORTHOGONAL SETS OF FUNCTIONS 39
3.3 STURM–LIOUVILLE PROBLEMS: ORTHOGONAL
FUNCTIONS
We now proceed to show that solutions of a certain ordinary differential equation with certain
boundary conditions (called a Sturm–Liouville problem) are orthogonal functions with respect to a
weighting function, and that therefore a well-behaved function can be represented by an infinite
series of these orthogonal functions (called eigenfunctions), as in Eqs. (3.12) and (3.16).
Recall that the problem
2
X xx + λ X = 0, X(0) = 0, X(1) = 0 0 ≤ x ≤ 1 (3.40)
has solutions only for λ = nπ and that the solutions, sin(nπx) are orthogonal on the interval
(0, 1). The sine functions are called eigenfunctions and λ = nπ are called eigenvalues.
As another example, consider the problem

2
X xx + λ X = 0 (3.41)
with boundary conditions

X(0) = 0
(3.42)
X(1) + HX x (1) = 0
The solution of the differential equation is

X = A sin(λx) + B cos (λx)) (3.43)

The first boundary condition guarantees that B = 0. The second boundary condition is satisfied
by the equation

A[sin(λ) + Hλ cos(λ)] = 0 (3.44)
Since A cannot be zero, this implies that

− tan(λ) = Hλ. (3.45)

The eigenfunctions are sin(λx)and the eigenvalues are solutions of Eq. (3.45). This is illustrated
graphically in Fig. 3.4.
We will generally be interested in the fairly general linear second-order differential
equation and boundary conditions given in Eqs. (3.46) and (3.47).
d dX
r(x) + [q(x) + λp(x)]X = 0 a ≤ x ≤ b (3.46)
dx dx
a 1 X(a) + a 2 dX(a)/dx = 0

b 1 X(b) + b 2 dX(b)/dx = 0 (3.47)