# Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (5th Edition)

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## Book Preface

This text discusses partial differential equations in the engineering and physical sciences. It is suited for courses whose titles include â€œFourier series,â€ â€œorthogonal functions,â€ or â€œboundary value problems.â€ It may also be used in courses on Greenâ€™s functions, transform methods, or portions on advanced engineering mathematics and mathematical methods in the physical sciences. It is appropriate as an introduction to applied mathematics.

Simple models (heat flow, vibrating strings, and membranes) are emphasized. Equations are formulated carefully from physical principles, motivating most mathematical topics. Solution techniques are developed patiently. Mathematical results frequently are given physical interpretations. Proofs of theorems (if given at all) are presented after explanations based on illustrative examples. Over 1000 exercises of varying difficulty form an essential part of this text. Answers are provided for those exercises marked with a star (âˆ—). Further details concerning the solutions for most of the starred exercises are available in an instructorâ€™s manual available for download through the Instructor Resource Center at PearsonHigherEd.com.

Standard topics such as the method of separation of variables, Fourier series, orthogonal functions, and Fourier transforms are developed with considerable detail. Finite difference numerical methods for partial differential equations are clearly presented with considerable depth. A briefer presentation is made of the finite element method. This text also has an extensive presentation of the method of characteristics for linear and nonlinear wave equations, including discussion of the dynamics of shock waves for traffic flow. Nonhomogeneous problems are carefully introduced, including Greenâ€™s functions for Laplaceâ€™s, heat, and wave equations. Numerous topics are included, such as differentiation and integration of Fourier series, Sturmâ€“Liouville and multidimensional eigenfunctions, Rayleigh quotient, Bessel functions for a vibrating circular membrane, and Legendre polynomials for spherical problems. Some optional advanced material is included (for example, asymptotic expansion of large eigenvalues, calculation of perturbed frequencies using the Fredholm alternative, stability conditions for finite difference methods, and direct and inverse scattering).

Applications briefly discussed include the lift and drag associated with fluid flow past a circular cylinder, Snellâ€™s law of refraction for light and sound waves, the derivation of the eikonal equation from the wave equation, dispersion relations for water waves, wave guides, and fiber optics.

The text has evolved from the authorâ€™s experiences teaching this material to different types of students at various institutions (MIT, UCSD, Rutgers, Ohio State, and Southern Methodist University). Prerequisites for the reader are calculus and elementary ordinary differential equations. (These are occasionally reviewed in the text, where necessary.) For the beginning student, the core material for a typical course consists of most of Chapters 1â€“5 and 7. This will usually be supplemented by a few other topics.

The text is somewhat flexible for an instructor, since most of Chapters 6â€“13 depend only on Chapters 1â€“5. Chapter 11 on Greenâ€™s functions for the heat and wave equation is an exception, since it requires Chapters 9 and 10.

Chapter 14 is more advanced, discussing linear and nonlinear dispersive waves, stability, and perturbation methods. It is self-contained and accessible to strong undergraduates. Group velocity and envelope equations for linear dispersive waves, whose applications include the rainbow caustic of optics, are analyzed. Nonlinear dispersive waves are discussed, including an introductory presentation of solitons for the weakly nonlinear long wave equation (Kortewegâ€“de Vries) and the weakly nonlinear wave envelope equation (nonlinear Schrodinger). In addition, instability and bifurcation phenomena for partial differential equations are discussed, along with perturbation methods (multiple scale and boundary layer problems). In Chapter 14, I have attempted to show the vitality of the contemporary study of partial differential equations in the context of physical problems.

The book also includes diffusion of a chemical pollutant, Galerkin numerical approximation for the frequencies, similarity solution for the heat equation, two-dimensional Greenâ€™s function for the wave equation, nonuniqueness of shock velocity and its resolution, spatial structure of traveling shock wave, stability and bifurcation theory for systems of ordinary differential equations, two-spatial-dimensional wave envelope equations, analysis of modulational instability, long wave instabilities, pattern formation for reaction diffusion equations, and the Turing instability.

NEW TO THIS EDITION: In the fifth edition, I have made an effort to preserve the fourth edition while making significant improvements. A simple and improved presentation of the linearity principle is done in Chapter 2, showing that the heat equation is a linear equation. Chapter 4 contains a straightforward derivation of the vibrating membrane, a great improvement over previous editions. I have added a few new, simpler exercises. Many of the exercises in which partial differential equations are solved in Chapters 2, 4, 5, 7, and 10 have been simplified by adding substantial hints, the core of a typical first course. The questions are usually the same, so that users of previous editions will not have difficulty adjusting to the new edition. In these exercises, often the hint includes the separation of variables itself, so the problem is more straightforward for the student. The student should obtain the correct answer to the initial and boundary value problems for partial differential equations more easily.

TECHNOLOGY: There are over 200 figures to illustrate various concepts, which I prepared using matlab. The matlab m-files for most of the mathematical figures may be obtained from my web page: http://faculty.smu.edu/rhaberma. Modern technology is especially important in its graphical ability, and I have tried to indicate throughout the text places where three-dimensional visualization is helpful. Overall, my object has been to explain clearly many fundamental aspects of partial differential equations as an introduction to this vast and important field. After achieving a certain degree of competence and understanding, the student can use this text as a reference, but for additional information the reader should be prepared to refer to other books such as the ones cited in the Bibliography.

Finally, it is hoped that this text enables the reader to find enjoyment in the study of the relationships between mathematics and the physical sciences