Introduction to Numerical Ordinary and Partial Differential Equations Using MATLAB

834 Pages·2004·37.41 MB·English

by  Alexander Stanoyevitch

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Introduction to Numerical
Ordinary and Partial
Differential Equations
Using MATLAB®
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
Introduction to Numerical
Ordinary and Partial
Differential Equations
Using MATLAB®
Alexander Stanoyevitch
WILEY-
INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Stanoyevitch, Alexander
Introduction to numerical ordinary and partial differential equations using MATLAB*
Alexander Stanoyevitch.
p. cm.
Includes bibliographical references and index.
ISBN 0-471-69738-9 (cloth : acid-free paper)
1. Differential equations—Numerical solutions—Data processing. 2. Differential
equations, Partial—Numerical solutions—Data processing. 3. MATLAB. I. Title.
QA371.5.D37.S78 2005
515V352—dc22 2004058042
Printed in the United States of America
10 9 8 7 6 5 4 3 21
Contents
Preface ix
PART I: Introduction to MATLAB and
Numerical Preliminaries
Chapter 1: MATLAB Basics 1
Section 1.1: What Is MATLAB?
Section 1.2: Starting and Ending a MATLAB Session
Section 1.3: A First MATLAB Tutorial
Section 1.4: Vectors and an Introduction to MATLAB Graphics
Section 1.5: A Tutorial Introduction to Recursion on MATLAB
Chapter 2: Basic Concepts of Numerical Analysis 23
with Taylor's Theorem
Section 2.1: What Is Numerical Analysis?
Section 2.2: Taylor Polynomials
Section 2.3: Taylor's Theorem
Chapter 3: Introduction to M-Files 45
Section 3.1: What Are M-files?
Section 3.2: Creating an M-file for a Mathematical Function
Chapter 4: Programming in MATLAB 57
Section 4.1: Some Basic Logic
Section 4.2: Logical Control Flow in MATLAB
Section 4.3: Writing Good Programs
Chapter 5: Floating Point Arithmetic and Error Analysis 85
Section 5.1: Floating Point Numbers
Section 5.2: Floating Point Arithmetic: The Basics
♦Section 5.3:1 Floating Point Arithmetic: Further Examples and Details
Chapter 6: Rootfinding 107
Section 6.1: A Brief Account of the History of Rootfinding
1 An asterisk that precedes a section indicates that the section may be skipped without a significant loss
of continuity to the main development of the text.
v
VI Contents
Section 6.2: The Bisection Method
Section 6.3: Newton's Method
♦Section 6.4: The Secant Method
♦Section 6.5: Error Analysis and Comparison of Rootfmding Methods
Chapter 7: Matrices and Linear Systems 143
Section 7.1: Matrix Operations and Manipulations with MATLAB
♦Section 7.2: Introduction to Computer Graphics and Animation
Section 7.3: Notations and Concepts of Linear Systems
Section 7.4: Solving General Linear Systems with MATLAB
Section 7.5: Gaussian Elimination, Pivoting, and LU Factorization
♦Section 7.6: Vector and Matrix Norms, Error Analysis, and Eigendata
Section 7.7: Iterative Methods
PART II: Ordinary Differential Equations
Chapter 8: Introduction to Differential Equations 285
Section 8.1: What Are Differential Equations?
Section 8.2: Some Basic Differential Equation Models and Euler's
Method
Section 8.3: More Accurate Methods for Initial Value Problems
♦Section 8.4: Theory and Error Analysis for Initial Value Problems
♦Section 8.5: Adaptive, Multistep, and Other Numerical Methods for
Initial Value Problems
Chapter 9: Systems of First-Order Differential Equations 355
and Higher-Order Differential Equations
Section 9.1: Notation and Relations
Section 9.2: Two-Dimensional First-Order Systems
Section 9.3: Phase-Plane Analysis for Autonomous First-Order Systems
Section 9.4: General First-Order Systems and Higher-Order Differential
Equations
Chapter 10: Boundary Value Problems for Ordinary 399
Differential Equations
Section 10.1: What Are Boundary Value Problems and How Can They
Be Numerically Solved?
Section 10.2: The Linear Shooting Method
Section 10.3: The Nonlinear Shooting Method
Section 10.4: The Finite Difference Method for Linear BVPs
♦Section 10.5: Rayleigh-Ritz Methods
Contents vii
PART III: Partial Differential Equations
Chapter 11: Introduction to Partial Differential Equations 459
Section 11.1: Three-Dimensional Graphics with MATLAB
Section 11.2: Examples and Concepts of Partial Differential Equations
Section 11.3: Finite Difference Methods for Elliptic Equations
Section 11.4: General Boundary Conditions for Elliptic Problems and
Block Matrix Formulations
Chapter 12: Hyperbolic and Parabolic Partial Differential 523
Equations
Section 12.1: Examples and Concepts of Hyperbolic PDEs
Section 12.2: Finite Difference Methods for Hyperbolic PDEs
Section 12.3: Finite Difference Methods for Parabolic PDEs
Chapter 13: The Finite Element Method 599
Section 13.1: A Nontechnical Overview of the Finite Element Method
Section 13.2: Two-Dimensional Mesh Generation and Basis Functions
Section 13.3: The Finite Element Method for Elliptic PDEs
Appendix A: Introduction to MATLAB's Symbolic 691
Toolbox
Appendix B: Solutions to All Exercises for the Reader 701
References 799
MATLAB Command Index 805
General Index 809
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