NUMERICALSOLUTIONOF ORDINARYDIFFERENTIAL …

1 NUMERICAL SOLUTION OFORDINARY DIFFERENTIALEQUATIONSK endall Atkinson, Weimin Han, David StewartUniversity of IowaIowa City, IowaA JOHN WILEY & SONS, INC., PUBLICATIONC opyrightc 2009 by John Wiley & Sons, Inc. All rights by John Wiley & Sons, Inc., Hoboken, New simultaneously in part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee tothe Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400,fax (978) 646-8600, or on the web at Requests to the Publisher for permission shouldbe addressed to the Permissions Department, John Wiley & Sons, Inc.

2 , 111 River Street, Hoboken, NJ07030, (201) 748-6011, fax (201) of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts inpreparing this book, they make no representations or warranties with respect to the accuracy orcompleteness of the contents of this book and specifically disclaim any implied warranties ofmerchantability or fitness for a particular purpose. No warranty may be created ore extended by salesrepresentatives or written sales materials. The advice andstrategies contained herin may not besuitable for your situation. You should consult with a professional where appropriate. Neither thepublisher nor author shall be liable for any loss of profit or any other commercial damages, includingbut not limited to special, incidental, consequential, or other general information on our other products and services please contact our Customer CareDepartment with the at 877-762-2974, outside the at 317-572-3993 or fax also publishes its books in a variety of electronic formats.

3 Some content that appears in print,however, may not be available in electronic 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 ????????????? (pbk.)1. Numerical analysis. 2. Ordinary differential Atkinson, Kendall E. II. is a trademark of The MathWorks, Inc. and is used with MathWorks does not warrant the accuracy of the text or exercises in this book s use or discussion of MATLABR software or related products does notconstitute endorsement or sponsorship by The MathWorks of aparticular pedagogicalapproach or particular use of the MATLABR in the United States of 9 8 7 6 5 4 3 2 1To Alice, Huidi, and SuePrefaceThis book is an expanded version of supplementary notes thatwe used for a course onordinary differential equations for upper-division undergraduate students and begin-ning graduate students in mathematics, engineering, and sciences.

4 The book intro-duces the numerical analysis of differential equations, describing the mathematicalbackground for understanding numerical methods and givinginformation on whatto expect when using them. As a reason for studying numericalmethods as a partof a more general course on differential equations, many of the basic ideas of thenumerical analysis of differential equations are tied closely to theoretical behaviorassociated with the problem being solved. For example, the criteria for the stabilityof a numerical method is closely connected to the stability of the differential equationproblem being book can be used for a one-semester course on the numerical solution of dif-ferential equations, or it can be used as a supplementary text for a course on the theoryand application of differential equations.

5 In the latter case, we present more aboutnumerical methods than would ordinarily be covered in a class on ordinary differentialequations. This allows the instructor some latitude in choosing what to include, andit allows the students to read further into 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 anintroductory text on the numerical solution of differential also include MATLABR programs to illustrate many of the ideas that areintroduced in the text. Much is to be learned by experimenting with the numericalsolution of differential equations. The programs in the book can be downloaded fromthe following site also contains graphical user interfaces for use inexperimenting with Euler smethod and the backward Euler method.

6 These are to be used from within theframework of methods vary in their behavior, and the many different types of differ-ential equation problems affect the performance of numerical methods in a variety ofways. An excellent book for real world examples of solvingdifferential equationsis that of Shampine, Gladwell, and Thompson [74].The authors would like to thank Olaf Hansen, California State University at SanMarcos, for his comments on reading an early version of the book. We also expressour appreciation to John Wiley of differential equations: An solvability of the initial value fields11 Problems132 Euler s of Euler s analysis of Euler s error Richardson Rounding error accumulation30 Problems32ixxCONTENTS3 Systems of differential differential methods for systems42 Problems464 The backward Euler method and the trapezoidal backward Euler trapezoidal method56 Problems625 Taylor and Runge Kutta Kutta A general framework for explicit Runge Kutta , stability.

7 And asymptotic Error prediction and Kutta Fehlberg Runge Kutta Two-point collocation methods87 Problems896 Multistep Bashforth Moulton MATLAB ODE codes105 Problems1067 General error analysis for multistep general error Stability Convergence Relative stability and weak stability122 Problems123 CONTENTSxi8 Stiff differential method of lines for a parabolic MATLAB programs for the method of differentiation regions for multistep sources of A-stability and Time-varying problems and the finite-difference codes146 Problems1479 Implicit RK methods for stiff differential of implicit Runge Kutta of Runge Kutta Kutta methods for stiff equations in practice160 Problems16110 Differential algebraic Initial conditions and DAEs as stiff differential Numerical issues: higher index Backward differentiation methods for Index 1 Index 2 Runge Kutta methods for Index 1 Index 2 Index three problems from Runge Kutta methods for mechanical index 3 Higher index DAEs184 Problems18511 Two-point boundary value A finite-difference A numerical Boundary conditions involving the Nonlinear two-point boundary value Finite difference Shooting Collocation Other methods and problems206 Problems20612 Volterra integral Solvability Special Numerical The trapezoidal Error for the trapezoidal General schema for numerical Numerical methods : Numerical Practical numerical stability227 Problems231 Appendix A.

8 Taylor s Theorem235 Appendix B. Polynomial interpolation241 References245 Index250 IntroductionDifferential equations are among the most important mathematical tools used in pro-ducing models in the physical sciences, biological sciences, and engineering. In thistext, we consider numerical methods for solving ordinary differential equations, thatis, those differential equations that have only one independent differential equations we consider in most of the book are of the formY (t) =f(t,Y(t)),whereY(t)is an unknown function that is being sought. The given functionf(t,y)of two variables defines the differential equation, and examples are given in Chapter1. This equation is called afirst-order differential equationbecause it contains afirst-order derivative of the unknown function, but no higher-order derivative.

9 Thenumerical methods for a first-order equation can be extendedin a straightforward wayto a system of first-order equations. Moreover, a higher-order differential equationcan be reformulated as a system of first-order brief discussion of the solvability theory of the initial value problem for ordi-nary differential equations is given in Chapter 1, where theconcept of stability ofdifferential equations is also introduced. The simplest numerical method,Euler smethod, is studied in Chapter 2. It is not an efficient numerical method, but it is anintuitive way to introduce many important ideas. Higher-order equations and systemsof first-order equations are considered in Chapter 3, and Euler s method is extended12 INTRODUCTIONto such equations. In Chapter 4, we discuss some numerical methods with betternumerical stability for practical computation.

10 Chapters 5and 6 cover more sophisti-cated and rapidly convergent methods , namely Runge Kutta methods and the familiesof Adams Bashforth and Adams Moulton methods , respectively. In Chapter 7, wegive a general treatment of the theory of multistep numerical methods . The numericalanalysis of stiff differential equations is introduced in several early chapters, and itis explored at greater length in Chapters 8 and 9. In Chapter 10, we introduce thestudy and numerical solution of differential algebraic equations, applying some of theearlier material on stiff differential equations. In Chapter 11, we consider numericalmethods for solving boundary value problems of second-order ordinary differentialequations. The final chapter, Chapter 12, gives an introduction to the numerical solu-tion of Volterra integral equations of the second kind, extending ideas introduced inearlier chapters for solving initial value problems.