NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS PDF

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NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

Kendall Atkinson, Weimin Han, David Stewart

University of Iowa

Iowa City, Iowa

A JOHN WILEY & SONS, INC., PUBLICATION

Published simultaneously in Canada.

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however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Numerical Solution of Ordinary Differential Equations / Kendall E. Atkinson . . . [et al.]. p. cm.-(Wiley series in ??) "Wiley-Interscience."

Includes bibliographical references and index.

ISBN ??? (pbk.)

1. Numerical analysis. 2. Ordinary differential equations.

I. Atkinson, Kendall E. II. Series.

MATLABR?is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book's use or discussion of MATLABR?software or related products does not constitute endorsement or sponsorship by The MathWorks of aparticular pedagogical approach or particular use of the MATLABR?software.

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Preface

ordinary differential equations for upper-division undergraduatestudents and begin- ning graduate students in mathematics, engineering, and sciences. The book intro- duces the numerical analysis of differential equations, describing the mathematical background for understanding numerical methods and givinginformation on what to expect when using them. As a reason for studying numericalmethods as a part of a more general course on differential equations, many of the basic ideas of the numerical analysis of differential equations are tied closely to theoretical behavior associated with the problem being solved. For example, the criteria for the stability ofa numericalmethodis closelyconnectedtothe stabilityofthedifferentialequation problem being solved. This book can be used for a one-semester course on the numerical solution of dif- and application of differential equations. In the latter case, we present more about equations. This allows the instructor some latitude in choosing what to include, and it allows the students to read furtherinto topics that may interest them. For example, the book discusses methods for solving differential algebraic equations (Chapter 10) and Volterra integral equations (Chapter 12), topics not commonly included in an introductory text on the numerical solution of differential equations. vii viiiPREFACE

We also include MATLAB

R?programs to illustrate many of the ideas that are introduced in the text. Much is to be learned by experimenting with the numerical solutionofdifferentialequations. Theprogramsinthebookcanbedownloadedfrom the following website. This site also containsgraphicaluserinterfaces foruse inexperimentingwith Euler's method and the backward Euler method. These are to be used from within the framework of MATLAB. Numerical methods vary in their behavior, and the many different types of differ- ential equationproblemsaffect the performanceof numericalmethods in a variety of ways. An excellent book for "real world" examples of solvingdifferential equations is that of Shampine, Gladwell, and Thompson [74]. The authors would like to thank Olaf Hansen, California State University at San Marcos, for his comments on reading an early version of the book. We also express our appreciation to John Wiley Publishers.

Introduction1

1 Theory of differential equations: An introduction 3

1.1 General solvability theory7

1.2 Stability of the initial value problem 8

1.3 Direction fields11

Problems13

2 Euler's method15

2.1 Definition of Euler's method16

2.2 Error analysis of Euler's method 21

2.3 Asymptotic error analysis26

2.3.1 Richardson extrapolation 28

2.4 Numerical stability29

2.4.1 Rounding error accumulation 30

Problems32

ix xCONTENTS

3 Systems of differential equations 37

3.1 Higher-order differential equations 39

3.2 Numerical methods for systems 42

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