Kovach L.D. Boundary-Value Problems

Addison Wesley Publishing Company, 1984. — 426 p. — ISBN: 0-201-11728-2.The author's objective in writing Boundary-Value Problems is to present methods of solving second-order linear partial differential equations that arise in applications. For the most part, the methods Consist of separation of variables and (Laplace and Fourier) transform methods. Since these methods lead to ordinary differential equations, it is assumed that the student has the necessary background in this subject.Chapter 1 provides a ready reference to topics in ordinary differential equations and may be consulted as needed. ft is not intended that this reference chapter be included in a course in boundary-value problems, although the section on uniform convergence may be particularly helpful.
In Chapter 2 the difference between initial-value problems and boundary-value problems in ordinary differential equations is pointed out. This leads naturally into the prolific Sturm—Liouville theory, including the representation of a function by a series of orthonormal functions. Convergence and completeness are also presented here.
Separation of variables is introduced in Chapter 3 in connection with Laplace's equation and the heat equation. D'Alembert's solution of the vibrating infinite string is included and extended to a finite string with boundary conditions. The chapter concludes with the canonical forms of elliptic, parabolic, and hyperbolic equations.
Fourier series is the main topic in Chapter 4. In addition to cosine, sine, and exponential series, there is a section on applications and one on convergence.
A heuristic argument extends Fourier series to Fourier integrals in Chapter 5. This extension then leads naturally to the Fourier transform and its applications.
All of the foregoing is brought together in Chapter 6, which is devoted to the solution of boundary-value problems expressed in rectangular coordinates.
Included is the treatment of nonhomogeneous equations and nonhomogeneous boundary conditions. Fourier and Laplace transform methods of solution follow and there is also a Section on the important, but often neglected, topic of the verification of solutions.
Chapter 7 contains the necessary variations to boundary-value problems phrased in polar, cylindrical, and spherical coordinates. This leads into discussions of Bessel functions and Legendre polynomials and their properties.
A final chapter is devoted to numerical methods for solving boundary-value problems in both rectangular and polar coordinates.There is a conscious effort to separate theory, applications, and numerical methods. This is done not only to preserve the mainstream of the development without distracting ramifications but also to provide the instructor with a maximum amount of flexibility. Since some sections are independent of the main theme, they may be included or omitted depending on the available time and the makeup of the class.
Additional flexibility can be found in the exercises. There are more than 1400 exercises at the ends of sections and these are divided into three categories.
The first group helps to clarify the textual material and fills in details that have of necessity been omitted. The next, and most numerous, group consists of variations of the examples in the text and sufficient additional exercises to provide the practice most students need for a complete understanding of the material.
A third group Consists of more challenging exercises and those that extend the theory. Some suggestions for outside reading are also included in this group. Exercises that are of a computational nature are indicated by an asterisk (*).
It is a firm conviction of the author that the inclusion of historical sidelights is helpful to the reader and provides a welcome change of pace as well. Unfortunately, historical "facts" are sometimes controversial and provide fuel for hot debates on various topics, such as who was the originator of a particular method and the like. Unless otherwise indicated, we have used Florian Cajori's, A History of Mathematics, 3d ed. (New York: Chelsea, 1980) as a reference. This classic, first published in 1893, has endured to serve as an excellent source book.
Much of the material in this text has been used in classrooms during the past fifteen years. Some sections have appeared in the author's Advanced Engineering Mathematics (Reading, Mass.: Addison-Wesley, 1982). Preliminary versions of the present text were read by Ronald Guenther, Oregon State University; Roman Voronka, New Jersey Institute of Technology; and Euel Kennedy, California Polytechnic State University, and their incisive suggestions have been incorporated. The contributions of these reviewers is hereby gratefully acknowledged. Thanks are also due to Judy Caswell who did all the typing and to Wayne Yuhasz for his encouragement and editorial guidance.