Page 24 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

14 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
differential equation is

2
ξξ =−λ (2.10)
and we deduce that the two homogeneous boundary conditions are

(0) = 0
(2.11)
ξ (1) = 0

Solving the differential equation we find

= A cos(λξ) + B sin(λξ) (2.12)

where A and B are constants to be determined. The first boundary condition requires that
A = 0.
The second boundary condition requires that either B = 0orcos(λ) = 0. Since the former
cannot be true (U is not zero!) the latter must be true. ξ can take on any of an infinite number
of values λ n = (2n − 1)π/2, where n is an integer between negative and positive infinity.
Equation (2.10) together with boundary conditions (2.11) is called a Sturm–Liouville problem.
The solutions are called eigenfunctions and the λ n are called eigenvalues. A full discussion of
Sturm–Liouville theory will be presented in Chapter 3.
Hence the apparent solution to our partial differential equation is any one of the following:
2
2
U n = B n exp[−(2n − 1) π τ/4)] sin[π(2n − 1)ξ/2]. (2.13)
2.1.3 Superposition
Linear differential equations possess the important property that if each solution U n satisfies
the differential equation and the boundary conditions then the linear combination

∞ ∞

2 2
B n exp[−(2n − 1) π τ/4] sin[π(2n − 1)ξ/2] = U n (2.14)
n=1 n=1
also satisfies them, as stated in Theorem 2. Can we build this into a solution that satisfies the one
remaining boundary condition? The final condition (the nonhomogeneous initial condition)
states that
∞

1 = B n sin(π(2n − 1)ξ/2) (2.15)
n=1
This is called a Fourier sine series representation of 1. The topic of Fourier series is further discussed in
Chapter 3.