Description
Birkhauser Introduction to Partial Differential Equations with MATLAB by Jeffery M. Cooper
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._x000D_ Table of contents :- _x000D_
Chapter 1. Preliminaries _x000D_
Chapter 2. First-Order Equations: _x000D_
Method of characteristics for linear equations; nonlinear conservation laws; weak solutions; shock waves; numerical methods. Chapter 3. Diffusion: _x000D_
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: _x000D_
Separation of variables; eigenfunction expansions; symmetric boundary conditions; long-time behavior. Chapter 5. Waves Again: _x000D_
Gas dynamics; the nonlinear string; linearized model;_x000D_
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: _x000D_
Fourier series; Fourier transform and the heat equation; discrete Fourier transform; fast Fourier transform. Chapter 7. Dispersive Waves and the Schrodinger Equation: _x000D_
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: _x000D_
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: _x000D_
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: _x000D_
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_x000D_