2nd Edition

An Introduction to Partial Differential Equations with MATLAB

By Matthew P. Coleman Copyright 2013
    684 Pages 116 B/W Illustrations
    by Chapman & Hall

    An Introduction to Partial Differential Equations with MATLAB®, Second Edition illustrates the usefulness of PDEs through numerous applications and helps students appreciate the beauty of the underlying mathematics. Updated throughout, this second edition of a bestseller shows students how PDEs can model diverse problems, including the flow of heat, the propagation of sound waves, the spread of algae along the ocean’s surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom.

    Suitable for a two-semester introduction to PDEs and Fourier series for mathematics, physics, and engineering students, the text teaches the equations based on method of solution. It provides both physical and mathematical motivation as much as possible. The author treats problems in one spatial dimension before dealing with those in higher dimensions. He covers PDEs on bounded domains and then on unbounded domains, introducing students to Fourier series early on in the text.

    Each chapter’s prelude explains what and why material is to be covered and considers the material in a historical setting. The text also contains many exercises, including standard ones and graphical problems using MATLAB. While the book can be used without MATLAB, instructors and students are encouraged to take advantage of MATLAB’s excellent graphics capabilities. The MATLAB code used to generate the tables and figures is available in an appendix and on the author’s website.

    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
    Properties of Sine and Cosine
    The Fourier Series
    The Fourier Series, Continued
    The Fourier Series—Proof of Pointwise Convergence
    Fourier Sine and Cosine Series

    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

    First-Order PDEs with Constant Coefficients
    First-Order PDEs with Variable Coefficients
    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 Functions for Equations of Evolution

    Numerical Methods
    Finite Difference Approximations for ODEs
    Finite Difference Approximations for PDEs
    Spectral Methods and the Finite Element Method

    Appendix A: Uniform Convergence; Differentiation and Integration of Fourier Series
    Appendix B: Other Important Theorems
    Appendix C: Existence and Uniqueness Theorems
    Appendix D: A Menagerie of PDEs
    Appendix E: MATLAB Code for Figures and Exercises
    Appendix F: Answers to Selected Exercises




    Matthew P. Coleman

    "This is an excellent textbook … first, the book can be used by a person who has no interest in MATLAB at all, and, second, this book deserves to be considered by—in fact, should be at the top of the list of—any professor looking for an undergraduate text in PDEs. … there are several reasons why I view this book as being in the upper echelon of undergraduate PDE textbooks. One is the extremely high quality of exposition. Coleman writes clearly and cleanly, with a conversational tone and a high regard for motivation. He clearly has a great deal of experience teaching this subject and has learned what points are likely to cause confusion and therefore need expanded discussion. The author also employs the nice pedagogical feature of page-long ‘preludes’ to each chapter, which not only summarize what the chapter will cover and how it fits into the general theme of things, but also typically provide some brief historical commentary as well. In general, the overall effect of this book is like listening to a discussion by a good professor in office hours. … very highly recommended. I don’t know when or if I will ever teach an undergraduate PDE course, but if I ever do, this book will certainly be on my short list of possible texts."
    —Mark Hunacek, MAA Reviews, September 2013

    "… a pick for any college-level collection strong in applied mathematics and nonlinear science, and provides a thorough assessment updated for the latest mathematical applications. From modeling problems ranging from heat flow to sound waves and algae spread to equations based on methods of solution and physical and mathematical applications, this reviews PDEs and their applications and is a pick for advanced math collections whose patrons have an basic knowledge of multivariable calculus and ODEs. Any working with MATLAB codes and problem-solving applications need this!"
    —California Bookwatch, November 2013

    Praise for the First Edition:
    "The strongest aspect of this text is the very large number of worked boundary value problem examples."

    "This is a useful introductory text on PDEs for advanced undergraduate / beginning graduate students of applied mathematics, physics, or engineering sciences. … 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

    "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