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Separable Boundary-Value Problems in Physics

Separable Boundary-Value Problems in Physics

Morten Willatzen, Lok C. Lew Yan Voon

ISBN: 978-3-527-63493-4

May 2011

398 pages



Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level.

Separable Boundary-Value Problems in Physics is an accessible and comprehensive treatment of partial differential equations in mathematical physics in a variety of coordinate systems and geometry and their solutions, including a differential geometric formulation, using the method of separation of variables. With problems and modern examples from the fields of nano-technology and other areas of physics.

The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access.

The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians.

Part I Preliminaries
1. Introduction
2. General Theory
Part II Two-Dimensional Coordinate Systems
3. Rectangular Coordinates
4. Circular Coordinates
5. Elliptic Coordinates
6. Parabolic Coordinates
Part III Three-Dimensional Coordinate Systems
7. Rectangular Coordinates
8. Circular Cylinder Coordinates
9. Elliptic Cylinder Coordinates
10. Parabolic Cylinder Coordinates
11. Spherical Polar Coordinates
12. Prolate Spheroidal Coordinates
13. Oblate Spheroidal Coordinates
14. Parabolic Rotational Coordinates
15. Conical Coordinates
16. Ellipsoidal Coordinates
17. Paraboloidal Coordinates
Part IV Advanced Formulations
18. Differential Geometric Formulations
19. Quantum-mechanical Particle Confined to Neighborhood of Curves
20. Quantum-mechanical Particle Confined to Surfaces of Revolution
21. Boundary Perturbation Theory
A Hypergeometric Functions
B Baer Functions
C Bessel Functions
D Lamé Functions
E Legendre Functions
F Mathieu Functions
G Spheroidal Wave Functions
H Weber Functions
I Elliptic Integrals and Functions

“To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Thus this book can serve as a reference text for advanced undergraduate students.” (Zentralblatt MATH, 2012)

“They also consider a range of special functions that can result, though they make clear this is not a text on special functions. The fundamental material is suitable for a one-semester course in partial differential equations at the graduate or senior level; the rest of the material can be used as self-study or more advanced courses.”  (Book News, 1 April 2012)

"The text is clear and direct with respect to the mathematics and physics presented. Summing Up: Recommended. Upper-division undergraduates, graduate students, and researchers/faculty." (Choice, 1 December 2011)