Page 51 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

ORTHOGONAL SETS OF FUNCTIONS 41
Thus
b

(λ n − λ m ) p(x)X n X m dx = 0, m = n (3.53)
a
Notice that X m and X n are orthogonal with respect to the weighting function p(x) on the interval
(a, b). Obvious examples are the sine and cosine functions.

Example 3.3. Example 2.1 in Chapter 2 is an example in which the eigenfunctions are sin(λ n ξ)
and the eigenvalues are (2n − 1)π/2.

Example 3.4. If the boundary conditions in Example 2.1 in Chapter 2 are changed to

(0) = 0 (1) = 0 (3.54)

we note that the general solution of the differential equation is
(ξ) = A cos(λξ) + B sin(λξ) (3.55)

The boundary conditions require that B = 0and cos(λ) = 0. The values of λ can take on
any of the values π/2, 3π/2, 5π/2,..., (2n − 1)π/2. The eigenfunctions are cos(λ n ξ)and the
eigenvalue are λ n = (2n − 1)π/2.

Example 3.5. Suppose the boundary conditions in the original problem (Example 1, Chapter
2) take on the more complicated form

(0) = 0 (1) + h (1) = 0 (3.56)
The first boundary condition requires that B = 0. The second boundary conditions require that

sin(λ n ) + hλ n cos(λ n ) = 0, or (3.57)

1
λ n =− tan(λ n ) (3.58)
h
which is a transcendental equation that must be solved for the eigenvalues.The eigenfunctions
are, of course, sin(λ n x).

Example 3.6. A Physical Example: Heat Conduction in Cylindrical Coordinates

The heat conduction equation in cylindrical coordinates is
2
∂u ∂ u 1 ∂u
= + 0 < r < 1 (3.59)
∂t ∂r 2 r ∂r
with boundary conditions at R = 0and r = 1 and initial condition u(0,r) = f (r).