Numerical Solution of Ordinary Differential Equations PDF
Preview Numerical Solution of Ordinary Differential Equations
Numerical Solution of Ordinary Differential Equations Numerical Solution of Ordinary Differential Equations L. FOX AND D.F. MAYERS Oxford University LONDON NEW YORK CHAPMAN AND HALL First published in 1987 by Chapman and Hall Ltd 11 New FeUer Lane, London EC4P 4EE Published in the USA by Chapman and Hall 29 West 35th Street, New York NY 10001 © 1987 Fox and Mayers by J.W. Arrowsmith Ltd., Bristol All rights reserved. No part of this book may be reprinted, or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Fox, L. Numerical solution of ordinary differential equations. 1. Differential equations-Numerical solutions-Data processing I. TitJe H. Mayers, D.F. 515.3'5 QA370 Library of Congress Cataloging in Publication Data Fox, L. (Leslie) Numerical solution of ordinary differential equations. Bibliography: p. Includes index. 1. Differential equations-Numerical solutions. I. Mayers, D.F. (David Francis), 1931- . H. TitJe. QA372.F69 1987 515.3'5 87-5191 ISBN-13: 978-94-010-7907-5 e-ISBN-13: 978-94-009-3129-9 001: 10.1007/978-94-009-3129-9 Contents Preface IX 1 Introduction 1 1.1 Differential equations and associated conditions 1 1.2 Linear and non-linear differential equations 4 1.3 Uniqueness ofsolutions 6 1.4 Mathematical and numerical methods of solution 7 1.5 Difference equations 9 1.6 Additional notes 11 Exercises 12 2 Sensitivity analysis: inherent instability 14 2.1 Introduction 14 2.2 A simple example of sensitivity analysis 16 2.3 Variational equations 18 2.4 Inherent instability of linear recurrence relations. Initial- value problems 20 2.5 Inherent instability oflinear differential equations. Initial- value problems 27 2.6 Inherent instability: boundary-value problems 29 2.7 Additional notes 33 Exercises 39 3 Initial-value problems: one-step methods 42 3.1 Introduction 42 3.2 Three possible one-stepmethods(finite-difference methods) 43 3.3 Error analysis: linear problems 44 3.4 Error analysis and techniques for non-linear problems 48 3.5 Induced instability: partial instability 51 3.6 Systems ofequations 56 3.7 Improving the accuracy 58 vi Contents 3.8 More accurate one-step methods 62 3.9 Additional notes 69 Exercises 75 4 Initial-value problems: multi-step methods 78 4.1 Introduction 78 4.2 Multi-step finite-difference formulae 78 4.3 Convergence, consistency and zero stability 80 4.4 Partial and other stabilities 83 4.5 Predictor-corrector methods 88 4.6 Error estimation and choice of interval 90 4.7 Starting the computation 91 4.8 Changing the interval 92 4.9 Additional notes 94 Exercises 95 5 Initial-value methods for boundary-value problems 98 5.1 Introduction 98 5.2 The shooting method: linear problems 99 5.3 The shooting method: non-linear problems 101 5.4 The shooting method: eigenvalue problems 103 5.5 The shooting method: problems with unknown boundaries106 5.6 Induced instabilities of shooting methods 108 5.7 Avoiding induced instabilities 113 5.8 Invariant embedding for linear problems 117 5.9 Additional notes . 121 Exercises 125 6 Global (finite-difference) methods for boundary-value problems 128 6.1 Introduction 128 6.2 Solving linear algebraic equations 129 6.3 Linear differential equations oforders two and four 135 6.4 Simultaneous linear differential equations offirst order 141 6.5 Convenience and accuracy ofmethods 143 6.6 Improvement ofaccuracy 144 6.7 Non-linear problems 154 6.8 Continuation for non-linear problems 157 6.9 Additional notes 160 Exercises 174 Contents vii 7 Expansion methods 179 7.1 Introduction 179 7.2 Properties and computational importance of Chebyshev polynomials 179 7.3 Chebyshev solution ofordinary differential equations 181 7.4 Spline solution of boundary-value problems 189 7.5 Additional notes 192 Exercises 195 8 Algorithms 198 8.1 Introduction 198 8.2 Routines for initial-value problems 199 8.3 Routines for boundary-value problems 207 9 Further notes and bibliography 212 10 Answers to selected exercises 231 Index 247 Preface Nearly 20 years ago we produced a treatise (ofabout the same length as this book)entitled Computing methodsfor scientistsandengineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence ofthe 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety ofcomputational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized booksoneach major topic: for example on numerical linearalgebra, numerical approximation,numericalsolutionofordinarydifferentialequations,numerical solution ofpartial differential equations, and so on. These are needed because ournumericaleducationand software have improved and because our relevant problems exhibit more variety and more difficulty. Ordinary differential equa tionsareobviouscandidatesfor such treatment,and thecurrent book is written in this sense. Some mathematics in this sort ofwriting is ofcourse essential, and we were somewhat surprised 20 years ago to hear of professors of engineering being excited by the title of our book but alarmed by the standard ofmathematics needed for its understanding. There, and here, little more is needed than the ability to differentiateordinarilyand partially, to know alittleabout the Taylor series, and a little elementary material on matrix algebra in the solution of simultaneous linear algebraic equations and the determination ofeigenvalues andeigenvectorsofmatrices. Suchknowledge issurelyneeded byanybody with differentialequations to solve,and surelyno apologyis neededfor the inclusion of some mathematics at this level. If occasionally the mathematics is a little more difficult the reader should take it as read, and proceed confidently to the next section. Though there are two main types ofordinary differential equation, those of initial-value type and those of boundary-value type, most books until quite recentlyhaveconcentratedontheformer,andagainuntilrecentlytheboundary- x Preface value problem has had little literature. Very few books, even now, treat both problems between the same two covers, and this book, which does this in quite an elementaryway, must be almost unique in these respects. Aftera simple introduction there is quite a substantialchapteron the sensiti vity or the degree of'condition' of both types of problem, mainly with respect tothecorrespondingdifferenceequationsbutalso,analogouslyandmorebriefly, with respect to differential equations. We believe that this feature is very important butlargelyneglected,and that itisdesirable to know(and to practise that knowledge) how to determine the effects ofsmall changes in the relevant data on the answers, and how, ifnecessary, some reformulation ofthe original problem can reduce the sensitivity. Next there are two chapters on the initial value problem solved by one-step and multistep finite difference methods, the Taylor series method and Runge-Kutta methods, with much concentration on thepossibilityofinducedinstability(instabilityofthemethod)whichcanproduce very poor results even for well-conditioned problems. Two other chapters spend some 75 pages on the discussion and analysis of twomainmethodsforboundary-valueproblems.Thefirst usestechniqueswhich produceresultsfromacombinationofinitial-valueproblemsolutions,butwhich can beguilty ofsignificant induced instability. The second uses global methods, including the solution of sets of algebraic equations, which give good results unless the problem is very ill-conditioned. This chapter also describes two methodsforimprovingtheaccuracyoffirstapproximatesolutionswhich,though mentioned briefly in the first initial-value chapter, are far more useful in the boundary-valuecase. Thereisalso somediscussion oftechniques for producing a good first approximation, usually very necessary for the successful iterative solution ofnonlinear boundary-value problems. Ashort chapter then discusses methods of polynomial approximation for both initial-value and boundary valueproblems,herewith virtually thesamenumerical techniquesin bothcases. Almost every technique in every chapter is illustrated with numerical examples, with special reference to inherent instability (problem sensitivity)and toinducedinstability(afunctionofthechosencomputationalmethod). Exercises for the reader are presented at the end ofeach chapter, and the final chapter gives solutions to a selected numberoftheseexercises, particularly those which involve some non-trivial computation. We recommend the reader to try both the worked examples and these exercises on a computer, using any relevant software (programs) available, or in some cases with routines written by the reader. Thecomputershouldnot beexpected to giveexactlyourquoted results, since these depend on the machine's word length, rounding rules, and so on. If, however, the general shape of the solution does not agree with ours it is likely that the reader has made a mistake. In the eighth chapter we give a short account ofthe nature of the routines which are likely to be available currently in the best international program libraries. For the initial-value problem we also indicate some of the methods Preface xi which are used to produce solutions with guaranteed accuracy at each mesh point,theproductionofeconomicrelevantmethodsbeingoneoftheoutstanding problems in this field. At present there is far more numerical software for initial-value problems compared with boundary-value problems, and in our treatmentofthelatterwementionsomepointsnotyettreatedinlibraryroutines, including for example the difficulties and some possible techniques for dealing with problems with singularities or similar difficult behaviour at boundary points, perhaps particularly with eigenvalue problems. The ninth chaptermentions other problems which were treated rather briefly in the main text, or not at all, but its main content is a bibliography of some sixtybooksand papers,withcommentariesoneachentryrelevant to itscontent and its mathematical level. Most readers are unlikely to read all of these but the specific problems ofsome scientific disciplines may be eased by a study of some of the relevant literature. The final piece of advice we would give to a scientificorengineeringreaderistousehismathematicsorconsultaprofessional mathematician (and even a numerical analyst!) before starting any serious computation. Some qualitative knowledge about the nature ofthe solutioncan make a significant improvement in the choice of the best and most economic relevant computing method and program, on both the time taken and the accuracy ofthe results. Due to various causes and illnesses we have taken quite a long time in the writing of this book. We are very grateful for the close attention of a reader, selected by the publishers, who reported rather many weaknesses in the first draft on its style, ambiguities, topics perhaps wrongly included or omitted, and even a few errors. In our eyes the final version is a significant improvement. We also thank Chapman and Hall for the patience they have showed as the writingtimegotlongerandlonger,andforthequickapplicationofthepublishing sequence when they finally received the typescript. L. Fox D.F. Mayers May 1987
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