1st Edition

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

    860 Pages
    by A K Peters/CRC Press

    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.

    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

    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

    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