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
Boundary conditions |
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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 .