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An Introduction to Partial Differential Equations with MATLAB

Matthew P. Coleman
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Chapman & Hall/CRC Applied Mathematics & Nonlinear Science
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

This is an excellent textbook that may not, I fear, be particularly well served by its title. The phrase “with MATLAB” that appears there may suggest to a prospective user of this book that MATLAB is an essential part of the exposition, and thus might scare away any person who does not wish to use that software as part of a course on the subject. This would be unfortunate for two reasons: 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.

To deal with the former point first: MATLAB is used in this book in two principal ways. Diagrams and tables have been constructed using this software, and a number of exercises (carefully marked) require MATLAB for their solution. However, neither MATLAB nor any other software is required to understand any explanation in this book; a person using this book with nothing but pencil and paper at hand would get a great deal out of it. And there are so many exercises in the book that a reader can simply ignore any of the MATLAB exercises and still have a more than adequate sample for any class. So, while MATLAB definitely serves to enhance the discussion here, it is by no means necessary for somebody using this book.

As for the second point above, 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.

In addition, as in any good textbook, there are, as noted above, lots of exercises, covering a reasonable range of difficulty, although most seemed to be of the routine computational variety. A 20-page Appendix gives answers (generally without accompanying computation) to a selection of these. In addition, the publisher’s webpage for the book references a solutions manual for instructors who have adopted the text, but an email to the publisher resulted in the information that this manual will not be available until the end of October.

In keeping with the general student-friendly tone of the book, strict mathematical rigor is sometimes sacrificed in favor of readability. The reader will, for example, sometimes see a reference for a proof rather than the proof itself. For an elementary undergraduate course, this seems entirely appropriate. For a graduate course in mathematics, though, this point may be more troublesome (although graduate students in other disciplines who want to know how to use partial differential equations rather than prove things about them may find much here of interest).

Another feature of the book that I like (except perhaps for one small point, about which I am somewhat conflicted) is its organization. I came to this book with no formal training at all in PDEs; whatever I knew about them was largely self-taught, mostly learned from the (now long out of print) book Partial Differential Equations: An Introduction by Eutiqio Young. That book convinced me that the subject was attractive enough for me to look at other books over the years, and it quickly became apparent to me that there were several ways to organize the material.

Some books, for example, introduce Fourier series quite early (on page 17, for example, of Asmar’s Partial Differential Equations with Fourier Series and Boundary Value Problems) while others wait until the need for them has been more extensively motivated. Some books (like Haberman’s Applied Partial Differential Equations) plunge more or less immediately into a detailed study of one major example (in Haberman’s case, the heat equation); others do not.

Young’s book started with first-order PDEs (linear and quasi-linear), which was then followed by a discussion of linear equations and the classification of second-order linear PDEs with constant coefficients into three canonical forms. He then discussed in some detail the representatives of each form — first the wave equation, and then (after a chapter on Fourier series motivated by the wave equation) chapters on the heat equation and Laplace equation.

This seemed logical enough, but the book under review takes the different approach of looking not at each equation individually but instead considering solution methods as the motivating theme. (Having never taught a course in PDEs, I can’t recount from personal experience how successful this approach would be in a classroom, but it certainly reads well enough.)

The author begins with an excellent introductory chapter discussing basic terminology and examples and then examining very simple PDEs that can actually be solved from basic principles — for example, PDEs that are really ODEs in disguise. The method of separation of variables and eigenvalue problems, make their first appearance here as well.

The second chapter introduces the “big three” second-order linear PDEs with constant coefficients that were mentioned above. It is stated, but not yet shown, that these are the three canonical forms for the general linear second-order constant coefficient PDE. The emphasis in this chapter is on the physical derivation of each of these three equations, but the method of separation of variables is used to make a start at solving them, and this leads directly to Fourier series, the subject of chapter 3.

Fourier series, and separation of variables, are then applied in chapter 4 to discuss solutions of the big three PDEs on finite domains — i.e., with boundary conditions as well as initial conditions. First homogenous equations with homogenous boundary conditions are treated; then the boundary conditions are allowed to be non-homogenous, and finally the equations themselves are allowed to be non-homogenous.

The next chapter introduces the method of characteristic curves. Here again, the level of complexity gradually rises: the first equations considered are first order linear equations with constant coefficients (with the convection equation derived and used as a motivating example), followed by first order linear equations with variable coefficients. Quasi-linear equations are referred to in the exercises. Then, second order equations are considered. The first to be discussed is the wave equation on an infinite domain, solvable by a simple change of variable which results in the notion of characteristic lines. In the next section, these ideas are extended to study the wave equation on semi-infinite and finite domains, and then in the next section they are extended to the case of the general second-order linear PDE, first with constant coefficients and then (briefly) with general coefficients.

In chapter 6, the notion of Fourier transform is introduced and applied to study other second-order linear PDEs on unbounded domains. Distributions make their first appearance in this chapter, introduced first in a very formal way and then made more precise as linear functionals on the space of test functions.

These six chapters amount to what the author calls the “core” of the book, and corresponds to what might be covered in a one-semester course (or perhaps just a bit more). They only amount to about half of the text, however: the remaining chapters cover more advanced topics such as special functions and orthogonal polynomials, Sturm-Liouville theory, PDEs in higher dimensions, the Green’s function and numerical methods. There is no chapter dependence chart, but these chapters seem to be fairly independent of one another, and the author does suggest some ways in which parts of these chapters could be used to flesh out a one-semester course if time permits.

The one potential concern that I have with this arrangement of the material (at least as far as the first half of the book is concerned) is that the fairly late introduction of characteristic curves results in some topics that might be seen as candidates for early discussion being deferred until fairly late into the semester. For example, as noted above, linear first order PDEs with constant coefficients can be easily solved by a simple change of variable, yet don’t really get discussed here until after the arguably more complicated second-order PDEs are. Also, it can be readily observed early on that the wave equation (utt = cuxx) has the general solution f(x+ct) + g(x-ct), which in turn has a visually pleasing interpretation as the sum of two moving waves; since this is directly relevant to the theory of characteristic curves, it, too, is deferred until late in the semester. However, this does not seem like a terribly serious issue; people who are determined to introduce these topics earlier could easily modify the path through the book to accommodate this desire.

There is yet a third reason why I put this book high on the list of potential texts for an undergraduate course, and that is the reasonable price: prices fluctuate, of course, but as I write this, a new copy of this book can be obtained on for less than 72 dollars, a much more reasonable price than is charged there for either Strauss (almost $139) or Haberman ($129). More and more, I find myself thinking seriously about the price of a book when I select one for a course, and a disparity like this, it seems to me, should weigh reasonably heavily in favor of this book.

Verdict: 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 ([email protected]) teaches mathematics at Iowa State University. 

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