2nd Edition

# An Introduction to Partial Differential Equations with MATLAB

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**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.

**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

**Characteristics **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

*L*

^{2}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 PDEsAppendix E: MATLAB Code for Figures and Exercises Appendix F: Answers to Selected Exercises **

**References **

**Index**

### Biography

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."

—SIAM"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