Synopses & Reviews
The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB is a careful integration of traditional core topics with modern topics, taking full advantage of the computational power of MATLAB to enhance the learning experience. This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes. Topic and Features: • Nonlinear equations including nonlinear conservation laws; • Dispersive wave equations and the Schrodinger equation; • Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform; • Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site. This text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners.
Review
"Cooper's book stands out among a host of PDE works. It not only adequately treats traditional core partial differential equation methods but also integrates analytic solutions with numerical schemes through the implementation of MATLAB routines. As an application-oriented book that provides the basic definitions, theorems, and analyses of the solutions, it contains the core topics needed for a sound background in partial differential equations.... One of the book's excellent features is the availability of illustrative and challenging problems, some of which have been cast in the form of MATLAB projects. Such features undoubtedly make this a suitable work for a laboratory component of an introductory PDEs course. Recommended. Undergraduates through faculty." --Choice
Synopsis
Overview The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. The core consists of solution methods, mainly separation of variables, for boundary value problems with constant coeffi- cients in geometrically simple domains. Too often an introductory course focuses exclusively on these core problems and techniques and leaves the student with the impression that there is no more to the subject. Questions of existence, uniqueness, and well-posedness are ignored. In particular there is a lack of connection between the analytical side of the subject and the numerical side. Furthermore nonlinear problems are omitted because they are too hard to deal with analytically. Now, however, the availability of convenient, powerful computational software has made it possible to enlarge the scope of the introductory course. My goal in this text is to give the student a broader picture of the subject. In addition to the basic core subjects, I have included material on nonlinear problems and brief discussions of numerical methods. I feel that it is important for the student to see nonlinear problems and numerical methods at the beginning of the course, and not at the end when we run usually run out of time. Furthermore, numerical methods should be introduced for each equation as it is studied, not lumped together in a final chapter.
Table of Contents
Chapter 1. Preliminaries Chapter 2. First-Order Equations: Method of characteristics for linear equations; nonlinear conservation laws; weak solutions; shock waves; numerical methods. Chapter 3. Diffusion: Diffusion on the line; maximum principle; fundamental solution of the heat equation; Burgers' equation; numerical methods. Chapter 4. Boundary Value Problems for the Heat Equation: Separation of variables; eigenfunction expansions; symmetric boundary conditions; long-time behavior. Chapter 5. Waves Again: Gas dynamics; the nonlinear string; linearized model; the linear wave equation without boundaries; boundary value problems on the half-line and finite interval; conservation of energy;numerical methods; nonlinear Klein-Gordon equation. Chapter 6. Fourier Series and Fourier Transform: Fourier series; Fourier transform and the heat equation; discrete Fourier transform; fast Fourier transform. Chapter 7. Dispersive Waves and the Schrodinger Equation: Method of stationary phase; dispersive equation (group velocity and phase velocity); Schrodinger equation; spectrum of the Schrodinger operator. Chapter 8. The Heat and Wave Equations in Higher Dimensions: Fundamental solution of heat equation; eigenfunctions for the disk and rectangle; Kirchoff's formula for the wave equation; nodal curves; conservation of energy; the Maxwell equations. Chapter 9. Equilibrium: Harmonic functions; maximum principle; Dirichlet problem in the disk and rectangle; Poisson kernel; Green's functions; variational problems and weak solutions. Chapter 10. Numerical Methods in Higher Dimensions: Finite differences; finite elements; Galerkin methods, A reaction-diffusion equation. Chapter 11. Epilogue: Classification Appendix A: Recipes and Formulas Appendix B: Elements of MATLAB Appendix C: References Appendix D: Solutions to Selected Problems Appendix E: List of Computer Programs Index