1st Edition

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




ISBN 9781568813356
Published June 18, 2009 by A K Peters/CRC Press
860 Pages

USD $155.00

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Book Description

This book is a text on partial differential equations (PDEs) of mathematical physics and boundary value problems, trigonometric Fourier series, and special functions. This is the core content of many courses in the fields of engineering, physics, mathematics, and applied mathematics. The accompanying software provides a laboratory environment that allows the user to generate and model different physical situations and learn by experimentation. From this standpoint, the book along with the software can also be used as a reference book on PDEs, Fourier series and special functions for students and professionals alike.

Table of Contents

Fourier Series
Periodic Processes and Periodic Functions
Fourier Formulas
Orthogonal Systems of Functions
Convergence of Fourier Series
Fourier Series for Non-periodic Functions
Fourier Expansions on Intervals of Arbitrary Length
Fourier Series in Cosine or Sine Functions
The Complex Form of the Fourier Series
Complex Generalized Fourier Series
Fourier Series for Functions of Several Variables
Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Completeness of a System of Trigonometric Functions
General Systems of Functions: Parseval’s Equality and Completeness
Approximation of Functions in the Mean
Fourier Series of Functions Given at Discrete Points
Solution of Differential Equations by Using Fourier Series
Fourier Transforms
The Fourier Integral
Problems
Sturm-Liouville Theory
The Sturm-Liouville Problem
Mixed Boundary Conditions
Examples of Sturm-Liouville Problems
Problems
One-Dimensional Hyperbolic Equations
Derivation of the Basic Equations
Boundary and Initial Conditions
Other Boundary Value Problems: Longitudinal Vibrations of a Thin Rod
Torsional Oscillations of an Elastic Cylinder
Acoustic Waves
Waves in a Shallow Channel
Electrical Oscillations in a Circuit
Traveling Waves: D’Alembert Method
Semi-infinite String Oscillations and the Use of Symmetry Properties
Finite Intervals: The Fourier Method for One-Dimensional Wave Equations
Generalized Fourier Solutions
Energy of the String
Problems
Two-Dimensional Hyperbolic Equations
Derivation of the Equations of Motion
Oscillations of a Rectangular Membrane
The FourierMethod Applied to Small Transverse Oscillations of a Circular Membrane
Problems
One-Dimensional Parabolic Equations
Physical Problems Described by Parabolic Equations: Boundary Value Problems
The Principle of the Maximum, Correctness, and the Generalized Solution
The Fourier Method of Separation of Variables for the Heat Conduction Equation
Heat Conduction in an Infinite Bar
Heat Equation for a Semi-infinite Bar
Problems
Parabolic Equations for Higher-Dimensional Problems
Heat Conduction in More than One Dimension
Heat Conduction within a Finite Rectangular Domain
Heat Conduction within a Circular Domain
Problems
Elliptic Equations
Elliptic Partial Differential Equations and Related Physical Problems
The Dirichlet Boundary Value Problem for Laplace’s Equation in a Rectangular Domain
Laplace’s and Poisson’s Equations for Two-Dimensional Domains with Circular Symmetry
Laplace’s Equation in Cylindrical Coordinates
Problems
Bessel Functions
Boundary Value Problems Leading to Bessel Functions
Bessel Functions of the First Kind
Properties of Bessel Functions of the First Kind: Jn (x)
Bessel Functions of the Second Kind
Bessel Functions of the Third Kind
Modified Bessel Functions
The Effect of Boundaries on Bessel Functions
Orthogonality and Normalization of Bessel Functions
The Fourier-Bessel Series
Further Examples of Fourier-Bessel Series Expansions
Spherical Bessel Functions
The Gamma Function
Problems
Legendre Functions
Boundary Value Problems Leading to Legendre Polynomials
Generating Function for Legendre Polynomials
Recurrence Relations
Orthogonality of Legendre Polynomials
The Multipole Expansion in Electrostatics
Associated Legendre Functions P m (x) 
                                                     n
Orthogonality and the Normof Associated Legendre Functions
Fourier-Legendre Series in Legendre Polynomials
Fourier-Legendre Series in Associated Legendre Functions
Laplace’s Equation in Spherical Coordinates and Spherical Functions
Problems
Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem
Auxiliary Functions for Different Types of Boundary Conditions
The Sturm-Liouville Problem and the Laplace Equation
Vector Calculus
How to Use the Software Associated with this Book
Program Overview
Examples Using the Program TrigSeries
Examples Using the Program Waves
Examples Using the Program Heat
Examples Using the Program Laplace
Examples Using the Program FourierSeries

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Author(s)

Biography

Victor Henner, Department of Physics and Astronomy, University of Louisville, Kentucky, USA

Tatyana Belozerova, Perm State University, Russia

Kyle Forinash, Professor of Physics and Program Coordinator, Indiana University Southeast, New Albany, USA

Reviews

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 and Mehdi Hassani, MAA Reviews, November 2009

[Henner, Forinash, and Belozerova] address the main topics of many courses in mathematical physics within the fields of engineering, physics, mathematics, and applied mathematics. The texbook and accompanying software are significantly more detailed than typical introductions to partial differential equations, they say, and provide examples on setting up physical problems as mathematical ones, solving partial differential equations under different types of boundary conditions, working with special functions, and carrying out a Fourier analysis using these functions. The software provides a simple interface, and does not require students to learn a programming language. 
Book News Inc., September 2009