Page 25 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 15
2.1.4 Orthogonality
It may seem hopeless at this point when we see that we need to find an infinite number of
constants B n . What saves us is a concept called orthogonality (to be discussed in a more general
way in Chapter 3). The functions sin(π(2n − 1)ξ/2) form an orthogonal set on the interval
0 <ξ < 1, which means that

1

sin(π(2n − 1)ξ/2) sin(π(2m − 1)ξ/2)dξ = 0 when m = n (2.16)
0
= 1/2 when m = n

Hence if we multiply both sides of the final equation by sin(π(2m − 1)ξ/2)dξ and integrate
over the interval, we find that all of the terms in which m = n are zero, and we are left with
one term, the general term for the nth B, B n

1
4

B n = 2 sin(π(2n − 1)ξ/2)dξ = (2.17)
π(2n − 1)
0
Thus

∞
4
2 2
U = exp[−π (2n − 1) τ/4] sin[π(2n − 1)ξ/2] (2.18)
π(2n − 1)
n=1
satisfies both the partial differential equation and the boundary and initial conditions, and
therefore is a solution to the boundary value problem.

2.1.5 Lessons
We began by assuming a solution that was the product of two variables, each a function of only
one of the independent variables. Each of the resulting ordinary differential equations was then
solved. The two homogeneous boundary conditions were used to evaluate one of the constant
coefficients and the separation constant λ. It was found to have an infinite number of values.
These are called eigenvalues and the resulting functions sinλ n ξ are called eigenfunctions. Linear
superposition was then used to build a solution in the form of an infinite series. The infinite
series was then required to satisfy the initial condition, the only nonhomogeneous condition.
The coefficients of the series were determined using the concept of orthogonality stated in
Theorem 3, resulting in a Fourier series. Each of these concepts will be discussed further in
Chapter 3. For now we state that many important functions are members of orthogonal sets.