Page 526 - Advanced engineering mathematics

506 CHAPTER 15 Special Functions and Eigenfunction Expansions

which has the form

(ry ) + (q + λp)y = 0. (15.1)

When p,q,r, and r are continuous on (a,b), and r(x)> 0 and p(x)> 0on (a,b), we call

equation (15.1) the Sturm-Liouville differential equation.

This Sturm-Liouville equation contains a quantity λ. We want to determine values of λ,
called eigenvalues, such that there are nontrivial (not identically zero) solutions y of the differen-
tial equation which satisfy certain conditions at a and b. For a given eigenvalue λ, such a solution
is an eigenfunction associated with λ. Conditions at a and b that solutions must satisfy are called
boundary conditions. There are three kinds of boundary value problems for the eigenvalues and
eigenfunctions, depending on the form of the boundary conditions. In each problem, we assume
that p(x)> 0 and r(x)> 0on (a,b).

The Regular Sturm-Liouville Problem
We want numbers λ for which there are nontrivial solutions of equation (15.1) satisfying regular
boundary conditions

A 1 y(a) + A 2 y (a) = 0 and B 1 y(b) + B 2 y (b) = 0,
in which A 1 and A 2 are given numbers, not both zero, and B 1 and B 2 are also given numbers, not
both zero.
The Periodic Sturm-Liouville Problem
r(a) = r(b) and we want numbers λ for which there are nontrivial solutions of equation (15.1)
satisfying periodic boundary conditions

y(a) = y(b) and y (a) = y (b).
The Singular Sturm-Liouville Problem
r(a) = 0or r(b) = 0, but not both. For this problem we want numbers λ for which there are
nontrivial solutions of equation (15.1), subject to the following boundary conditions at a or b.
If r(a) = 0 then there is the single boundary condition
B 1 y(b) + B 2 y (b) = 0,

with B 1 and B 2 not both zero.
If r(b) = 0, then there is the single boundary condition
A 1 y(a) + A 2 y (a) = 0,

with A 1 and A 2 given and not both zero.
We will derive properties of eigenvalues and eigenfunctions after looking at two examples.

EXAMPLE 15.1 A Regular Sturm-Liouville Problem
The problem

y + λy = 0; y(0) = y(L) = 0
is regular on [0, L]. We will solve it for later use. Consider cases on λ.

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October 14, 2010 15:20 THM/NEIL Page-506 27410_15_ch15_p505-562