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Mathematics

Many problems in the physical world can be modeled by partial differential equations, from applications as diverse as the flow of heat, the vibration of a ball, the propagation of sound waves, the diffusion of ink in a glass of water, electric and magnetic fields, 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. However, as with any area of applied mathe­matics, the field of PDEs is interesting not only because of its applications, but because it has taken on a mathematical life of its own. The author has written this book with both ideas in mind, in the hope that the student will appreciate the usefulness of the subject and, at the same time, get a glimpse into the beauty of some of the underlying mathematics.

This text is suitable for a two-semester introduction to partial differen­tial equations and Fourier series for students who have had basic courses in multivariable calculus (through Green's and the Divergence Theorems) and ordinary differential equations. Over the years, the author has taught much of the material to undergraduate mathematics, physics and engineering students at Penn State and Fairfield Universities, as well as to beginning graduate en­gineering students at Penn State. It is assumed that the student has not had a course in real analysis. Thus, we treat pointwise convergence of Fourier series and do not talk about mean-square convergence until Chapter 8 (and, there, in terms of the Riemann, and not the Lebesgue, integral). Further, we feel that it is not appropriate to introduce so subtle an idea as uniform convergence in this setting, so we discuss it only in the Appendices.

One may approach the teaching of PDEs in one of two ways: either based on type of equation, or based on method of solution. While appreciating the importance of the former idea, we have chosen the latter approach, as it

1. allows us to treat problems in one spatial dimension before dealing with those in higher dimensions, and
2. allows the text to be used for a one-semester course without the need to jump around.

A typical one-semester course would cover the "core" Chapters 1-4, then any two of Chapters 5, 6, 7 and 11. If the class consists of mathematics majors only, then one need not dally in Chapter 2, thus allowing more flexibility. Further, one might choose to include some of the beginning material in Chap­ters 9 and/or 10 or, if the students have seen special functions previously, one may elect to do (for example) most of Chapter 9. In these latter cases, the instructor may need to supplement the material a bit, in order to preserve continuity of presentation.

Motivation

The author believes that it is essential to provide the students with motivation (other than grade!) for each of the various topics. We have tried, as far as possible, to provide such motivation, both physical and mathematical (so, for example, the Fourier series is introduced only after the need for it, through solving the heat equation via separation of variables, has been estab­lished). Further, we begin by considering PDEs on bounded domains—PDE boundary-value problems—before looking at unbounded domains, because

1. This approach allows us to get to Fourier series early on.
2. Problems on bounded domains are more natural than those on unbounded domains, at least in one dimension.

Further, and in this same vein, we have provided a Prelude to each chapter, the purpose of which is to describe the topics to be covered in the chapter, so as to tell the student what is coming and why it is coming, and to put the material into its historical setting, as well.

Exercises

Of course, mathematics is not a spectator sport, and can only be learned by doing. Thus, it goes without saying (but we're saying it anyway) that the exercises are a key part of the text. Basically, they are of four types:

1. "solve-the-problem" exercises,
2. proofs
3. extend the material exercises
4. graphical exercises.

Types (1) and (2) are self-explanatory. As for type (3), there are some topics that.we choose to present as exercises, for various reasons. In some cases, these will be problems that are similar enough to those already solved in the text. In others, they may involve material which is "important," but which is not necessary in later parts of the text. As some of these may be quite difficult, we make sure to lead the student through them when necessary. Alternatively, the instructor may choose to present the material herself in class.

Lastly, as PDEs is such a visual subject, we've provided a number of graph-ical exercises. Some of these can be done by hand, but the majority are to be performed using MATLAB (and these are labeled MATLAB, interestingly enough).

MATLAB

This text has been written so that it can be used without access to software. That said, it makes little sense to write a book on such a visual and intu­itive subject as PDEs without taking advantage of one of the multitude of mathematical software packages available these days. We have chosen MATLAB because it is, by far, the most user-friendly of the packages we've tried, because of its excellent graphics capabilities, and because it seems to be the software-of-choice among the engineering community (while making strong inroads in math and physics, as well).

This text does not pretend to be an introduction to MATLAB. There are a number of good books available for that purpose (for example, that by Sigmon and Davis listed in the Bibliography). For those wishing to use the MATLAB exercises, we assume that the student is familiar with the rudiments of the package—how to get it up-and-running, how M-files work, etc. What we have done is to use MATLAB to generate the tables and the more "mathematical" figures in the book, for which we've supplied the MATLAB code in Appendix E, and also on the Chapman & Hall/CRC Press website at www.crcpress.com/downloads.

Elementary Differential Equations by W. E. Kohler, Lee W. Johnson (Addison Wesley) This book is designed for the sophomore differential equations course taken by students majoring in science and engineering. We assume the reader has had a course in elementary calculus.

The authors have integrated the underlying theory, the solution procedures, and the numerical and computational aspects of differential equations as seamlessly as possible. For exa mple, when they introduce a new category of problems (such as first order equations, higher order equations, systems of differential equations, etc..), we begin with the basic existence-uniqueness theory since it forms the framework for understanding differential equations and attempting to solve them. However, because they want the text to be easy to read and understand, they discuss the theory as simply as possible and emphasize how to use it.

Linear and nonlinear equations (first order and higher order) are treated in separate chapters. They recognize there is a pedagogical trade off. On the one hand, order is a unifying characteristic of differential equations. On the other hand, linear differential equations are of such importance in terms of applications, theory, and solution techniques that they warrant a strong and separate emphasis. They have opted for the latter approach.

The theory of differential equations has an intrinsic beauty and provides an important tool for understanding the world around us. The interplay of math­ematics and science in helping to explain the physical world gives the subject of differential equations much of its vitality. Many readers studying differential equations are preparing for careers in the physical or life sciences. Therefore, when developing models, they try to guide the student carefully through the underlying physical principles leading to the relevant mathematics.

They also emphasize the importance of common sense, intuition, and "back­of-the-envelope" checks. It is important, when solving problems, for students to ask, "Does my answer make sense?" Some of the examples and exercises ask the student to anticipate and interpret the physical content of the solu­tion (for example, "Should we expect an equilibrium solution to exist in this application? If so, why? What should its value be?"). They feel it is particularly important to develop this type of mind-set in the present "computer age," when the temptation is great to accept any computer-generated output as correct.

Some features of this book should be noted:

The authors have made a determined effort to write a text that is relatively easy for students to read and understand. When they introduce a new topic, such as separable first order equations or the Wronskian, they are careful to give illustrative examples and enough detail so that students can follow the discussion.

The text includes a large collection of exercises ranging from routine drill exercises to interesting applications drawn from a number of different disciplines. It also includes quite a few exercises that ask students to wrestle with the underlying ideas rather than simply turn an algorithmic crank. For example, they may state that the solution of the initial value problem is given by and then ask the student to determine the constants.

Answer Key Solutions of the odd-numbered exercises are given at the end of the text.

Currently, high school courses as well as first-year college courses expose virtually every student to calculators and computers. In their major courses, science and engineering students routinely use computers and computational software. Some of the exercises in this book require some type of electronic computational aid. However, the computational exercises are de­signed to be generic, not linked to any particular machine or software.

The basic ideas underlying nu­merical methods and their use in applications are presented early (in Sec­tions 3.8 and 6.9) in the context of Euler's method. Chapter 9 builds upon this introduction, developing a comprehensive treatment of one-step methods such as Runge-Kutta methods.

There is an Extended Problem at the end of each chapter. Each of these problems can serve as a miniproject. In some cases, the problem brings the concepts introduced in the chapter to bear on a particular application. In other cases, the purpose of the extended problem is to expand the student's horizon, showing how the material in the chapter can be generalized. In certain applications, such as food preservation, the problem exposes students to the mathematical aspects of state-of-the-art work.

An Appendix on Matrix Theory Chapter 6 discusses systems of linear differen­tial equations, y' = A(t)y + g(t). Later, central ideas of systems are revisited in Chapter 7 (Laplace Transforms) and in Chapter 8 (Systems of Nonlinear Equa­tions). However, not all students have the same background in matrix theory. Therefore, they have included an Appendix on Matrix Theory that summarizes the results necessary for understanding systems.

The Student Solutions Manual gives detailed solutions of the odd-numbered exercises. The solutions manual also gives examples showing how to use computational software (such as Mathematica, Maple, and MAT LAB) as an aid to solving the types of problems they include in the exercises. A web site for the book (http://www.aw.com/kohler) gives examples showing how to use computational software to solve typical differential equations.

Differential Equations, Second Edition by Paul Blanchard, Robert L. Devaney, Glen R. Hall (Brooks/Cole) is a product of the now complete National Science Foundation Boston Uni­versity Differential Equations Project sponsored by the National Science Foundation (NSF Grant DUE‑9352833) and Boston University. The goal of that project was to re­think the traditional, sophomore‑level differential equations course. This textbook represents the most up-to-date integration of technology to the use and utility of learning differential equations.

Editor’s summary: The study of differential equations is a beautiful application of the ideas and techniques of calculus to our everyday lives. Indeed, it could be said that calculus was developed mainly so that the fundamental principles that govern many phenomena could be ex­pressed in the language of differential equations. Unfortunately, it was difficult to con­vey the beauty of the subject in the traditional first course on differential equations be­cause the number of equations that can be treated by analytic techniques is very limited. Consequently, the course tended to focus on techniques rather than on concepts.

This book is an outgrowth of our opinion that we are now able to effect a radical revision, and we approach our updated course with several goals in mind. First, the tra­ditional emphasis on specialized tricks and techniques for solving differential equations is no longer appropriate given the technology that is readily available. Second, many of the most important differential equations are nonlinear, and numerical and qualitative techniques are more effective than analytic techniques in this setting. Finally, the dif­ferential equations course is one of the few undergraduate courses where it is possible to give students a glimpse of the nature of contemporary mathematical research.

The Qualitative, Numeric, and Analytic Approaches

Accordingly, this book is a radical departure from the typical "cookbook" differential equations text. We have eliminated most of the specialized techniques for deriving formulas for solutions, and we have replaced them with topics that focus on the for­mulation of differential equations and the interpretation of their solutions. To obtain an understanding of the solutions, we generally attack a given equation from three differ­ent points of view.

One major approach we adopt is qualitative. We expect students to be able to vi­sualize differential equations and their solutions in many geometric ways. For example, we readily use slope fields, graphs of solutions, vector fields, and solution curves in the phase plane as tools to gain a better understanding of solutions. We also ask students to become adept at moving among these geometric representations and more traditional analytic representations.

Since differential equations are readily studied using the computer, we also em­phasize numerical techniques. The CD that accompanies this book provides students with ample computer‑based tools to investigate the behavior of solutions of differential equations both numerically and graphically. Even if we can find an explicit formula for a solution, we often work with the equation both numerically and qualitatively to un­derstand the geometry and the long‑term behavior of solutions. When we can find ex­plicit solutions easily (such as in the case of separable first‑order equations or constant-­coefficient, linear systems), we do the calculations. But we never fail to examine the formulas we obtain using qualitative and numerical points of view as well.

How This Book Is Different

There are several specific ways in which this book differs from other books at this level. First, we incorporate modeling throughout. We expect students to understand the meaning of the variables and parameters in a differential equation and to be able to interpret this meaning in terms of a particular model. Certain models reappear often as running themes and are drawn from a variety of disciplines so that students with various backgrounds will find something familiar.

We also adopt a dynamical systems point of view. Thus, we are always concerned with the long‑term behavior of solutions of an equation, and using all the appropriate approaches outlined above, we ask students to predict this long-term behavior of solutions. In addition, we emphasize the role of parameters in many of our examples, and we specifically address the manner in which the behavior of solutions changes as these parameters are varied.

A major new feature of this edition is the inclusion of a CD that contains a variety of computer‑based resources. These resources include solvers which allow the student to compute and display graphically numerical solutions of both first‑order and systems of differential equations. Also included are a number of demonstrations that allow students and teachers to investigate in detail specific topics covered in the text. Certain exercises refer directly to these computational resources.

We begin, as other texts, with first‑order equations. However, the only analytic technique we use to find closed‑form solutions is separation of variables (and, at the end of the chapter, an integrating factor or two to handle certain linear equations). Instead, we emphasize the meaning of a differential equation and its solutions in terms of its slope field and the graphs of its solutions. If the differential equation is autonomous, we also discuss its phase line. This discussion of the phase line serves as an elementary introduction to the idea of a phase plane, which plays a fundamental role in subsequent chapters.

We then move directly from first‑order equations to systems of first‑order differential equations. Rather than consider second‑order equations separately, we convert these equations to first‑order systems. When these equations are viewed as systems, we are able to use qualitative and numerical techniques more readily. Of course, we then use the information about these systems gleaned from these techniques to recover information about the solutions of the original equation.

We also begin the treatment of systems with a general approach. We do not immediately restrict our attention to linear systems. Qualitative and numerical techniques work just as easily when a system is nonlinear, and one can proceed a long way toward understanding systems without resorting to algebraic techniques. However, qualitative ideas do not tell the whole story, and we are led naturally to the idea of linearization. With this background in the fundamental geometric and qualitative concepts, we then discuss linear systems in detail. As always, we not only emphasize the formula for the general solution of a linear system, but also the geometry of its solution curves and their relationship to the eigen-values and eigenvectors.

While our study of systems requires the minimal use of some linear algebra, it is definitely not a prerequisite. Since we deal primarily with two‑dimensional systems, we easily develop all of the necessary algebraic techniques as we proceed. In the process, we give considerable insight into the geometry of such topics as eigenvectors and eigen-values.

These topics form the core of our approach. However, there are many additional topics that one would like to cover in the course. Consequently, we have included discussions of forced second‑order equations, nonlinear systems, Laplace transforms, numerical methods, and discrete dynamical systems. Although some of these topics are quite traditional, we always present them in a manner that is consistent with the philosophy developed in the first half of the text.

At the end of each chapter, we have included several "labs." Doing detailed nu­merical experimentation and writing reports has been our most successful modifica­tion of the traditional course at Boston University. Good labs are tough to write and to grade, but we feel that the benefit to students is extraordinary,

Changes in the Second Edition

To our knowledge, no new analytic techniques for solving the differential equations in this text have appeared since the first edition (or even in the last 100 years). However, advances in technology provide ample justification for a new edition. For example, recently developed magnetorheological fluids allow the damping coefficient of a har­monic oscillator to be adjusted in real time. These devices, which are just starting to appear in consumer goods, can be modeled and studied quite effectively using qualita­tive and numerical techniques (see Lab 2.5 and Section 4,4).

Perhaps the greatest change in this edition is the inclusion of a CD. The demon­strations on the CD are the equivalent of the type of in‑class demonstrations done in chemistry and physics lectures (except they don't smell bad or explode). The main difference is that the students can take these demonstrations home and repeat them. Ex­ercises have been included that refer specifically to the demonstrations. We have al­ways assumed that students using this text have access to general-purpose solvers like those on the CD. We hope that inclusion of these solvers with the text will circumvent problems in obtaining software and encourage students to use the numerical techniques from the very beginning of the course.