Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions

As the title of book indicates, this book explains mathematical methods and tools in solving some problems in physics. It doesn’t cover introductory mathematical topics in physics. The authors focus mainly on physics-related topics from partial differential equations (PDEs) and boundary value problems, Fourier series, and some special functions. The book has nine chapters, each including problems and exercises, some of them with solutions. A CD of software is attached to the book, which provides a laboratory environment that allows the user to generate and model different physical situations and learn by experimentation.

The first chapter is about Fourier series: the chapter explains how we can solve PDEs (mainly those that appeared in physical problems) by using Fourier series. The second chapter explains Sturm-Liouville theory, with examples, and the third chapter studies one-dimensional hyperbolic equations, which covers solution of various physical problems such as the oscillation of a string, acoustic waves, and electrical oscillation in a circuit. At the end of chapter, the authors solve the one-dimensional wave equation using Fourier methods.

Chapter four studies two-dimensional hyperbolic equations, with applications to two dimensional flexible surfaces called membranes, focusing only on rectangular and circular ones. The mathematical formulation of heat conduction and diffusion both lead to parabolic equations, which are the subject of chapters five and six. Chapter seven is about Elliptic PDEs. The authors give familiar examples such as the equations of Laplace and Poisson.

Bessel functions and Legendre functions can be used a set of basis functions for the Fourier expansions of solutions to certain types of physical problems. The aim of chapters eight and nine is to study these functions. The attached CD provides a simple interface, and does not require students to learn a programming language. Appendix E describes how the reader can use this package and gives examples.

In comparison with typical introductions to partial differential equations, the book and attached software are significantly more detailed. It explains various examples of physical problems and solves related partial differential equations under different types of boundary conditions. The authors do more with special functions and carry out examples of Fourier analysis using these functions.

The book, along with the software, can also be considered as a reference book on PDEs, Fourier series and some of the special functions for students and professionals. As a text, this book can be used in an advanced course on Mathematical Physics (or related courses) for advanced students of engineering, physics, mathematics, and applied mathematics.

Soheila Emamyari is a Ph.D. student in Physics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, where she is working on her thesis about polymers, under direction of Professor H. Fazli.

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics, Analytic Number Theory in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.