Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

Learn how to solve complex differential equations using MATLAB® Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB® teaches readers how to numerically solve both ordinary and partial differential equations with ease. This innovative publication brings together a skillful treatment of MATLAB and programming alongside theory and modeling. By presenting these topics in tandem, the author enables and encourages readers to perform their own computer experiments, leading them to a more profound understanding of differential equations. The text consists of three parts: Introduction to MATLAB and numerical preliminaries, which introduces readers to the software and itsgraphical capabilities and shows how to use it to write programs Ordinary Differential Equations Partial Differential Equations All the tools needed to master using MATLAB to solve differential equations are provided and include: "Exercises for the Reader" that range from routine computations to more advanced conceptual and theoretical questions (solutions appendix included) Illustrative examples, provided throughout the text, that demonstrate MATLAB's powerful ability to solve differential equations Explanations that are rigorous, yet written in a very accessible, user-friendly style Access to an FTP site that includes downloadable files of all the programs developed in the text This textbook can be tailored for courses in numerical differential equations and numerical analysis as well as traditional courses in ordinary and/or partial differential equations. All the material has been classroom-tested over the course of many years, with the result that any self-learner with an understanding of basic single-variable calculus can master this topic. Systematic use is made of MATLAB's superb graphical capabilities to display and analyze results. An extensive chapter on the finite element method covers enough practical aspects (including mesh generation) to enable the reader to numerically solve general elliptic boundary value problems. With its thorough coverage of analytic concepts, geometric concepts, programs and algorithms, and applications, this is an unsurpassed pedagogical tool. Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB®......Page 5 Contents......Page 7 Preface......Page 11 Section 1.1: What Is MATLAB?......Page 17 Section 1.2: Starting and Ending a MATLAB Session......Page 18 Section 1.3: A First MATLAB Tutorial......Page 19 Section 1.4: Vectors and an Introduction to MATLAB Graphics......Page 23 Section 1.5: A Tutorial Introduction to Recursion on MATLAB......Page 30 Section 2.1: What Is Numerical Analysis?......Page 39 Section 2.2: Taylor Polynomials......Page 41 Section 2.3: Taylor's Theorem......Page 50 Section 3.1: What Are M-files?......Page 61 Section 3.2: Creating an M-file for a Mathematical Function......Page 65 Section 4.1: Some Basic Logic......Page 73 Section 4.2: Logical Control Flow in MATLAB......Page 76 Section 4.3: Writing Good Programs......Page 89 Section 5.1: Floating Point Numbers......Page 101 Section 5.2: Floating Point Arithmetic: The Basics......Page 102 Section 5.3: Floating Point Arithmetic: Further Examples and Details......Page 112 Section 6.1: A Brief Account of the History of Rootfinding......Page 123 Section 6.2: The Bisection Method......Page 126 Section 6.3: Newton's Method......Page 134 Section 6.4: The Secant Method......Page 144 Section 6.5: Error Analysis and Comparison of Root finding Methods......Page 148 Section 7.1: Matrix Operations and Manipulations with MATLAB......Page 159 Section 7.2: Introduction to Computer Graphics and Animation......Page 173 Section 7.3: Notations and Concepts of Linear Systems......Page 202 Section 7.4: Solving General Linear Systems with MATLAB......Page 205 Section 7.5: Gaussian Elimination, Pivoting, and LU Factorization......Page 219 Section 7.6: Vector and Matrix Norms, Error Analysis, and Eigendata......Page 240 Section 7.7: Iterative Methods......Page 268 Section 8.1: What Are Differential Equations?......Page 301 Section 8.2: Some Basic Differential Equation Models and Euler's Method......Page 304 Section 8.3: More Accurate Methods for Initial Value Problems......Page 318 Section 8.4: Theory and Error Analysis for Initial Value Problems......Page 329 Section 8.5: Adaptive, Multistep, and Other Numerical Methods for Initial Value Problems......Page 342 Section 9.1: Notation and Relations......Page 371 Section 9.2: Two-Dimensional First-Order Systems......Page 374 Section 9.3: Phase-Plane Analysis for Autonomous First-Order Systems......Page 388 Section 9.4: General First-Order Systems and Higher-Order Differential Equations......Page 402 Section 10.1: What Are Boundary Value Problems and How Can They Be Numerically Solved?......Page 415 Section 10.2: The Linear Shooting Method......Page 419 Section 10.3: The Nonlinear Shooting Method......Page 427 Section 10.4: The Finite Difference Method for Linear BVPs......Page 434 Section 10.5: Rayleigh-Ritz Methods......Page 442 Section 11.1: Three-Dimensional Graphics with MATLAB......Page 475 Section 11.2: Examples and Concepts of Partial Differential Equations......Page 484 Section 11.3: Finite Difference Methods for Elliptic Equations......Page 495 Section 11.4: General Boundary Conditions for Elliptic Problems and Block Matrix Formulations......Page 516 Section 12.1: Examples and Concepts of Hyperbolic PDEs......Page 539 Section 12.2: Finite Difference Methods for Hyperbolic PDEs......Page 556 Section 12.3: Finite Difference Methods for Parabolic PDEs......Page 589 Section 13.1: A Nontechnical Overview of the Finite Element Method......Page 613 Section 13.2: Two-Dimensional Mesh Generation and Basis Functions......Page 618 Section 13.3: The Finite Element Method for Elliptic PDEs......Page 652 Appendix A: Introduction to MATLAB's Symbolic Toolbox......Page 705 Appendix B: Solutions to All Exercises for the Reader......Page 717 References......Page 815 MATLAB Command Index......Page 821 General Index......Page 825 "Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB teaches readers how to numerically solve both ordinary and partial differential equations with ease. This publication brings together a skillful treatment of MATLAB and programming alongside theory and modeling. By presenting these topics in tandem, the author enables and encourages readers to perform their own computer experiments, leading them to a more profound understanding of differential equations." "This textbook can be tailored for courses in numerical differential equations and numerical analysis as well as traditional courses in ordinary and/or partial differential equations. All the material has been classroom-tested over the course of many years, with the result that any self-learner with an understanding of basic single-variable calculus can master this topic."--Jacket