# 4: Sturm-Liouville Boundary Value Problems

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We have seen that trigonometric functions and special functions are the solutions of differential equations. These solutions give orthogonal sets of functions which can be used to represent functions in generalized Fourier series expansions. At the same time we would like to generalize the techniques we had first used to solve the heat equation in order to solve more general initial-boundary value problems. Namely, we use separation of variables to separate the given partial differential equation into a set of ordinary differential equations. A subset of those equations provide us with a set of boundary value problems whose eigenfunctions are useful in representing solutions of the partial differential equation. Hopefully, those solutions will form a useful basis in some function space.

A class of problems to which our previous examples belong are the Sturm-Liouville eigenvalue problems. These problems involve self-adjoint (differential) operators which play an important role in the spectral theory of linear operators and the existence of the eigenfunctions needed to solve the interesting physics problems described by the above initial-boundary value problems. In this section we will introduce the Sturm-Liouville eigenvalue problem as a general class of boundary value problems containing the Legendre and Bessel equations and supplying the theory needed to solve a variety of problems.

• 4.1: Sturm-Liouville Operators
In physics many problems arise in the form of boundary value problems involving second order ordinary differential equations. For example, we will explore the wave equation and the heat equation in three dimensions. Separating out the time dependence leads to a three dimensional boundary value problem in both cases. Further separation of variables leads to a set of boundary value problems involving second order ordinary differential equations.
• 4.2: Properties of Sturm-Liouville Eigenvalue Problems
There are several properties that can be proven for the (regular) Sturm-Liouville eigenvalue problem.
• 4.3: The Eigenfunction Expansion Method
In this section we solve the nonhomogeneous problem Ly=f using expansions over the basis of Sturm-Liouville eigenfunctions. We have seen that Sturm-Liouville eigenvalue problems have the requisite set of orthogonal eigenfunctions. In this section we will apply the eigenfunction expansion method to solve a particular nonhomogeneous boundary value problem.
• 4.4: Appendix- The Fredholm Alternative Theorem
Given that Ly=f , when can one expect to find a solution? Is it unique? These questions are answered by the Fredholm Alternative Theorem. This theorem occurs in many forms from a statement about solutions to systems of algebraic equations to solutions of boundary value problems and integral equations. The theorem comes in two parts, thus the term "alternative". Either the equation has exactly one solution for all f , or the equation has many solutions for some f ’s and none for the rest.
• 4.5: Problems

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