Sturm-Liouville problems arise naturally in solving technical problems in engineering, physics, and more recently in biology and the social sciences. These problems lead to eigenvalue problems for ordinary and partial differential equations. Sturm-Liouville Problems: Theory and Numerical Implementation addresses, in a unified way, the key issues that must be faced in science and engineering applications when separation of variables, variational methods, or other considerations lead to Sturm-Liouville eigenvalue problems and boundary value problems.
1 Setting the Stage
3 Integral Equations
4 Regular Sturm-Liouville Problems
5 Singular Sturm-Liouville Problems - I
6 Singular Sturm-Liouville Problems – II
7 Approximation of Eigenvalues and Eigenfunctions
8 Concluding Examples and Observations
A Mildly Singular Compound Kernels
B Iteration of Mildly Singular Kernels
C The Kellogg Conditions
This is a mathematically rigorous, comprehensive, self-contained treatment of the elegant Sturm-Liouville theory. The book sets the stage with a review of classical applications involving buckling, vibrations, heat conduction and calculus of variations.
The role of Green's functions is introduced. In order to make the discussion which follows self-contained, a relevant review of real analysis is presented. Although experienced readers could skip this part, it is pleasing to see how nicely such results are frequently used to present the theory and carry out the proofs of the theorems.
Chapters are presented on integral equations, operators and kernels, regular and singular Sturm-Liouville problems. A thorough treatment is given on oscillation, approximation and orthogonality of eigenfunctions for regular and singular problems. Interlacement results for the zeros of eigenfunctions are given.
While classical examples leading to Bessel functions, Legendre polynomials, etc. are described, it must be admitted that closed formulae for solutions are seldom available for arbitrary systems and therefore a chapter is dedicated to numerical methods for approximating solutions. For this purpose, a chapter is dedicated to the description of the shooting method for differential equations, and numerical examples are presented. Combining the numerical discussion with the excellent analysis discussion makes the book unique. The book is rounded out with a relevant bibliography.
-Gene Allgower, Colorado State University
Sturm-Liouville theory is part of the bedrock of classical applied mathematics and mathematical physics. Guenther and Lee provide a excellently motivated compendium of topics ranging from physical considerations (Euler buckling, vibrations, diffusion, etc.) to integral representations and equations, the role of function spaces for solutions, and the Sturm-Liouville systems themselves. Graduate and advanced undergraduate students in applied mathematics and the physical sciences will find the development natural and leisurely, balancing mathematical rigor with intuition. A pleasant read.
-John Crow, Blackbird Analytics LLC