An Introduction to Partial Differential Equations with MATLAB

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Features

  • Introduces essential PDE concepts including Fourier series, integral transforms, and Green’s functions
  • Introduces many important PDEs via their physical applications
  • Offers a prelude to each chapter that describes topics covered and places material in historical context
  • Provides many graphical exercises, also available online, that are solved using MATLAB
  • Summary

    An Introduction to Partial Differential Equations with MATLAB exposes the basic ideas critical to the study of PDEs-- characteristics, integral transforms, Green’s functions, and, most importantly, Fourier series and related topics. The author approaches the subject from a motivational perspective, detailing equations only after a need for them has been established. He uses MATLAB® software to solve exercises and to generate tables and figures. This volume includes examples of many important PDEs and their applications.

    The first chapter introduces PDEs and makes analogies to familiar ODE concepts, then strengthens the connection by exploring the method of separation of variables. Chapter 2 examines the “Big Three” PDEs-- the heat, wave, and Laplace equations, and is followed by  chapters explaining how these and other PDEs on finite intervals can be solved using the Fourier series for arbitrary initial and boundary conditions.

    Chapter 5 investigates characteristics for both first- and second-order linear PDEs, the latter revealing how the Big Three equations are important far beyond their original application to physical problems. The book extends the Fourier method to functions on unbounded domains, gives a brief introduction to distributions, then applies separation of variables to PDEs in higher dimensions, leading to the special funtions, including the orthogonal polynomials.

    Other topics include Sturm-Liouville problems, adjoint and self-adjoint problems, the application of Green’s functions to solving nonhomogeneous PDEs, and an examination of practical numerical methods used by engineers, including the finite difference, finite element, and spectral methods.

    Table of Contents

    Introduction


    What are Partial Differential Equations?
    PDEs We Can Already Solve
    Initial and Boundary Conditions
    Linear PDEs--Definitions
    Linear PDEs--The Principle of Superposition
    Separation of Variables for Linear, Homogeneous PDEs
    Eigenvalue Problems

    The Big Three PDEs


    Second-Order,  Linear, Homogeneous PDEs with Constant Coefficients
    The Heat Equation and Diffusion
    The Wave Equation and the Vibrating String
    Initial and Boundary Conditions for the Heat and Wave Equations
    Laplace's Equation--The Potential Equation
    Using Separation of Variables to Solve the Big Three PDEs

    Fourier Series


    Introduction
    Properties of Sine and Cosine
    The Fourier Series
    The Fourier Series, Continued
    The Fourier Series---Proof of Pointwise Convergence
    Fourier Sine and Cosine Series
    Completeness
    Solving the Big Three PDEs
    Solving the Homogeneous Heat Equation for a Finite Rod
    Solving the Homogeneous Wave Equation for a Finite String
    Solving the Homogeneous Laplace's Equation on a Rectangular
    Domain
    Nonhomogeneous Problems
    Characteristicsfor Linear PDEs
    First-Order PDEs with Constant Coefficients
    First-Order PDEs with Variable Coefficients
    D'Alembert's Solution for the Wave Equation--The Infinite
    String
    Characteristics for Semi-Infinite and Finite String Problems
    General Second-Order Linear PDEs and Characteristics

    Integral Transforms


    The Laplace Transform for PDEs
    Fourier Sine and Cosine Transforms
    The Fourier Transform
    The Infinite and Semi-Infinite Heat Equations
    Distributions, the Dirac Delta Function and Generalized Fourier
    Transforms
    Proof of the Fourier Integral Formula
    Bessel Functions and Orthogonal Polynomials
    The Special Functions and Their Differential Equations
    Ordinary Points and Power Series Solutions; Chebyshev, Hermite
    and Legendre Polynomials
    The Method of Frobenius; Laguerre Polynomials
    Interlude: The Gamma Function
    Bessel Functions
    Recap: A List of Properties of Bessel Functions and Orthogonal
    Polynomials
    Sturm-Liouville Theory and Generalized Fourier Series
    Sturm-Liouville Problems
    Regular and Periodic Sturm-Liouville Problems
    Singular Sturm-Liouville Problems; Self-Adjoint Problems
    The Mean-Square or L2 Norm and Convergence in the Mean
    Generalized Fourier Series; Parseval's Equality and Completeness
    PDEs in Higher Dimensions
    PDEs in Higher Dimensions: Examples and Derivations
    The Heat and Wave Equations on a Rectangle; Multiple Fourier
    Series
    Laplace's Equation in Polar Coordinates; Poisson's Integral
    Formula
    The Wave and Heat Equations in Polar Coordinates
    Problems in Spherical Coordinates
    The Infinite Wave Equation and Multiple Fourier Transforms
    Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator;
    Green's Identities for the Laplacian
    Nonhomogeneous Problems and Green's Functions
    Green's Functions for ODEs
    Green's Function and the Dirac Delta Function
    Green's Functions for Elliptic PDEs (I): Poisson's Equation in
    Two Dimensions
    Green's Functions for Elliptic PDEs (II): Poisson's Equation in
    Three Dimensions; the Helmholtz Equation
    Green's Function's for Equations of Evolution

    Numerical Methods


    Finite Difference Approximations for ODEs
    Finite Difference Approximations for PDEs
    Spectral Methods and the Finite Element Method
    References
    Uniform Convergence; Differentiation and Integration of Fourier Series
    Important Theorems: Limits, Derivatives, Integrals, Series, and Interchange of Operations
    Existence and Uniqueness Theorems
    A Menagerie of PDEs
    MATLAB Code for Figures and Exercises
    Answers to Selected Exercises

    Editorial Reviews

    “The strongest aspect of this text is the very large number of worked boundary value problem examples.”
    SIAM review

    “This is a useful introductory text on partial differential equations (PDEs) for advanced undergraduate / beginning graduate students of applied mathematics, physics, or engineering sciences. …It may be said that this is a nice introductory text which certainly is of great use in preparing and delivering courses.”
    —Zentralblatt MATH

    “Readers new to the subject will find Coleman’s appendix cataloguing important partial differential equations in their natural surroundings quite useful. …Coleman’s more explicit, extended style would probably allow its use as an advanced graduate or reference text for UK engineers or physicists.”
    —Times Higher Education

    “The book presents very useful material and can be used as a basic text for self-study of PDEs.”
    —EMS Newsletter, Sept., 2005

    “Each chapter is introduced by a ‘prelude’ that describes its content and gives historical background. Each section concludes with a set of exercises, many of which are marked ‘MATLAB’.”
    —CMS Notes, Volume 37, No. 2, March 2005


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