Sturm-Liouville Theory and its Applications

Abstract

This book is based on lecture notes which I have used over a number of years to teach a course on mathematical methods to senior undergraduate students of mathematics at King Saud University. The course is offered here as a prereq- uisite for taking partial differential equations in the final (fourth) year of the undergraduate program. It was initially designed to cover three main topics: special functions, Fourier series and integrals, and a brief sketch of the Sturm– Liouville problem and its solutions. Using separation of variables to solve a boundary-value problem for a second-order partial differential equation often leads to a Sturm–Liouville eigenvalue problem, and the solution set is likely to be a sequence of special functions, hence the relevance of these topics. Typi- cally, the solution of the partial differential equation can then be represented (pointwise) by a Fourier series or a Fourier integral, depending on whether the domain is finite or infinite. But it soon became clear that these “mathematical methods” could be developed into a more coherent and substantial course by presenting them within the more general Sturm–Liouville theory in L2. According to this theory, a linear second-order differential operator which is self-adjoint has an orthogonal sequence of eigenfunctions that spans L2. This immediately leads to the fundamental theorem of Fourier series in L2 as a special case in which the operator is simply d2/dx2. The other orthogonal functions of mathematical physics, such as the Legendre and Hermite polynomials or the Bessel functions, are similarly generated as eigenfunctions of particular differential operators. The result is a generalized version of the classical theory of Fourier series, which ties up the topics of the course mentioned above and provides a common theme for the book.