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 programsOrdinary Differential EquationsPartial Differential EquationsAll 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 equationsExplanations that are rigorous, yet written in a very accessible, user-friendly styleAccess to an FTP site that includes downloadable files of all the programs developed in the textThis 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.
Author(s): Alexander Stanoyevitch
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
Edition: 1
Publisher: Wiley-Interscience
Year: 2004
Language: English
Pages: 832
Tags: Математика;Вычислительная математика;
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
