University Mathematics Handbook (2015)

XI. Ordinary Differential Equations, or ODE

Chapter 5. Sturm-Liouville Eigenvalue Problem

5.1  Definition

a.  A second-order ODE

(*)

with boundary conditions

(**)

when are given real numbers and is parameter, is called a boundary problem or a Sturm-Liouville problem.

b.  The solution of (*), (**) is called an eigenfunction of Sturm-Liouville problem.

c.  A for which problem (*), (**) has a solution is called an eigenvalue of the problem.

d.  Let functions , , be real and continuous on , and be eigenfunctions of a Sturm-Liouville problem corresponding to different eigenvalues , , respectively, when , are continuous on . Then, and are orthogonal with respect to weight function , that is, ,

e.  If, in addition to the conditions of paragraph d, , for every , then there is a countable set of real eigenvalues

f.  The orthogonal system of all eigenfunctions of Sturm-Liouville problem (*) is complete (see X.13.6) in the space of piecewise continuous functions on .

g.  If function is a piecewise differentiable on , then, for every of the same interval, the expansion of into a series of the eigenfunctions of Sturm-Liouville problem is

This series converges to in the points of continuity and to in the points of discontinuity.

h.  If and are continuous and is piecewise continuous on , then , and the series is absolutely and uniformly convergent on .

For example, if are Legendre polynomials, then

,

5.2  Examples

a.  The equation

b.  Legendre equations (see 3.3), can also be written in the Sturm-Liouville form

In this case, , , and , . The fitting eigen functions are Legendre polynomial .