Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 7Sturm–Liouville Theory

Majority of the frequently encountered partial differential equations in physics and engineering can be solved by the method of separation of variables. This method helps us to reduce a second-order partial differential equation into a set of ordinary differential equations with some new parameters called the separation constants. We have seen that solutions of these equations with the appropriate boundary conditions have properties reminiscent of an eigenvalue problem. In this chapter, we study these properties systematically in terms of the Sturm–Liouville theory.

7.1 Self-Adjoint Differential Operators

We define a second-order linear differential operator as

7.1

where are real functions with the first derivatives continuous. In addition, in the open interval , does not vanish even though it could have zeroes at the end points. We now define the ...