Boundary and Eigenvalue Problems in Mathematical Physics

Hans Sagan

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486661322
• eBook ISBN: 9780486150925
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This well-known text uses a limited number of basic concepts and techniques — Hamilton's principle, the theory of the first variation and Bernoulli's separation method — to develop complete solutions to linear boundary value problems associated with second order partial differential equations such as the problems of the vibrating string, the vibrating membrane, and heat conduction. It is directed to advanced undergraduate and beginning graduate students in mathematics, applied mathematics, physics, and engineering who have completed a course in advanced calculus.
In the first three chapters, Professor Sagan introduces Hamilton's principle and the theory of the first variation; he then discusses the representation of the vibrating string, the vibrating membrane and heat conduction (without convection) by partial differential equations. Bernoulli's separation method and infinite series solutions of homogeneous boundary value problems are introduced as a means for solving these problems.
The next three chapters take up Fourier series, self-adjoint boundary value problems, Legendre polynomials, and Bessel functions. The concluding three chapters address the characterization of eigenvalues by a variational principle; spherical harmonics, and the solution of the Schroedinger equation for the hydrogen atom; and the nonhomogeneous boundary value problem. Professor Sagan concludes most sections of this excellent text with selected problems (solutions provided for even-numbered problems) to reinforce the reader's grasp of the theories and techniques presented.

Table of contents

• Cover
• Title Page
• Copyright page
• Preface
• Acknowledgments
• Contents
• Chapter I: Hamilton’s Principle and the Theory of the First Variation
• Chapter II: Representation of Some Physical Phenomena by Partial Differential Equations
• Chapter III: Theorems Related to Partial Differential Equations and Their Solutions
• Chapter IV: Fourier Series
• Chapter V: Self-Adjoint Boundary Value Problems
• Chapter VI: Legendre Polynomials and Bessel Functions
• Chapter VII: Characterization of Eigenvalues by a Variational Principle
• Chapter VIII: Spherical Harmonics
• Chapter IX: The Nonhomogeneous Boundary Value Problem
• Appendix: Key for Reference Symbols used in the Appendix
• Answers and Hints to Even Numbered Problems
• Index