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This page intentionally left blank Solving ODEs with MATLAB This book is for people who need to solve ordinary differential equations (ODEs), both initialvalueproblems(IVPs)andboundaryvalueproblems(BVPs)aswellasdelaydifferential equations(DDEs). Thesetopicsareusuallytaughtinseparatecoursesoflengthonesemes- tereach,butSolvingODEswithMATLABprovidesasoundtreatmentofallthreeinabout250 pages. Thechaptersoneachofthesetopicsbeginwithadiscussionof“thefactsoflife”for theproblem,mainlybymeansofexamples. Numericalmethodsfortheproblemarethende- veloped–butonlythemethodsmostwidelyused. Althoughthetreatmentofeachmethodis briefandtechnicalissuesareminimized,theissuesimportantinpracticeandforunderstand- ingthecodesarediscussed. Oftensolvingarealproblemismuchmorethanjustlearninghow tocallacode. Thelastpartofeachchapterisatutorialthatshowshowtosolveproblemsby meansofsmallbutrealisticexamples. AbouttheAuthors L.F.ShampineisClementsProfessorofAppliedMathematicsatSouthernMethodistUniver- sityinDallas,Texas. I.GladwellisProfessorofMathematicsatSouthernMethodistUniversityinDallas,Texas. S.ThompsonisProfessorof MathematicsatRadfordUniversityinRadford,Virginia. ThisbookdistillsdecadesofexperiencehelpingpeoplesolveODEs. Theauthorsaccumulated thisexperienceinindustrialandlaboratorysettingsthatincludeNAG(NumericalAlgorithms Group), Babcock and Wilcox Company, Oak Ridge National Laboratory, Sandia National Laboratories, and The MathWorks – as well as in academic settings that include the Uni- versityofManchester, RadfordUniversity, andSouthernMethodistUniversity. Theauthors havecontributedtothesubjectbypublishinghundredsofresearchpapers,writingoreditinga half-dozenbooks,editingleadingjournals,andwritingmathematicalsoftwarethatisinwide use. WithassociatesatTheMathWorks,Inc.,theywrotealltheprogramsforsolvingODEs inMatlab,programsthatarethefoundationofthisbook. Solving ODEs with M ATLAB L. F. SHAMPINE I. GLADWELL S.THOMPSON SouthernMethodistUniversity SouthernMethodistUniversity RadfordUniversity    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521824040 © L. F. Shampine, I. Gladwell, S. Thompson 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 - ---- eBook (EBL) - --- eBook (EBL) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page vii 1 Getting Started 1 1.1 Introduction 1 1.2 Existence,Uniqueness,andWell-Posedness 6 1.3 StandardForm 19 1.4 ControloftheError 27 1.5 QualitativeProperties 34 2 InitialValue Problems 39 2.1 Introduction 39 2.2 NumericalMethodsforIVPs 40 2.2.1 One-StepMethods 41 2.2.2 MethodswithMemory 57 2.3 SolvingIVPsinMatlab 81 2.3.1 EventLocation 92 2.3.2 ODEsInvolvingaMassMatrix 105 2.3.3 LargeSystemsandtheMethodof Lines 114 2.3.4 Singularities 127 3 BoundaryValue Problems 133 3.1 Introduction 133 3.2 BoundaryValueProblems 135 3.3 BoundaryConditions 138 3.3.1 BoundaryConditionsatSingularPoints 139 3.3.2 BoundaryConditionsatInfinity 146 v vi Contents 3.4 NumericalMethodsforBVPs 156 3.5 SolvingBVPsinMatlab 168 4 Delay Differential Equations 213 4.1 Introduction 213 4.2 DelayDifferentialEquations 214 4.3 NumericalMethodsforDDEs 217 4.4 SolvingDDEsinMatlab 221 4.5 OtherKindsofDDEsandSoftware 247 Bibliography 251 Index 257 Preface This book is for people who want to solve ordinary differential equations (ODEs), both initialvalueproblems(IVPs)andboundaryvalueproblems(BVPs)aswellasdelaydif- ferentialequations(DDEs). SolvingODEswithMATLABisatextforaone-semestercourse forupper-levelundergraduatesandbeginninggraduatestudentsinengineering,science, andmathematics. PrerequisitesareafirstcourseinthetheoryofODEsandasurveycourse in numerical analysis. Implicit in these prerequisites is some programming experience, preferablyinMatlab,andsomeelementarymatrixtheory. SolvingODEswithMATLABis alsoareferenceforprofessionalsinengineering,science,andmathematics. Withitthey can quickly obtain an understanding of the issues and see example problems solved in detail. Theycanusetheprogramssuppliedwiththebookastemplates. Itisusualtoteachthethreetopicsofthisbookatanadvancedlevelinseparatecourses ofonesemestereach. SolvingODEswithMATLABprovidesasoundtreatmentofallthree topics in about 250 pages. This is possible because of the focus and level of the treat- ment. ThebookopenswithachaptercalledGettingStarted. NextisachapteronIVPs. These two chapters must be studied in order, but the remaining two chapters (on BVPs andDDEs)areindependentofoneanother. Itiseasytocoveroneofthesechaptersina one-semestercourse,butthepreparationandsophisticationofthestudentswilldetermine whetheritispossibletodoboth. ThechapteronDDEscanbecoveredmorequicklythan the one on BVPs because only one approach is taken up and it is an extension of meth- odsstudiedinthechapteronIVPs. Eachchapterbeginswithadiscussionofthe“factsof life”fortheproblem,mainlybymeansofexamples. Numericalmethodsfortheproblem arethendeveloped–butonlythemethodsmostwidelyused. Althoughthetreatmentof eachmethodisbriefandtechnicalissuesareminimized,theissuesimportantinpractice arediscussed. Oftensolvingarealproblemismuchmorethanjustlearninghowtocalla code. Thelastpartofthechapterisatutorialthatshowshowtosolveproblemsbymeans ofsmallbutrealisticexamples. Althoughqualitysoftwareingeneralscientificcomputingisdiscussed,alltheexamples andexercisesaresolvedinMatlab. ThisismostadvantageousbecauseMatlab(2000) vii viii Preface hasbecomeanextremelyimportantproblem-solvingenvironment(PSE)forbothteach- ingandresearch. ThesolversofMatlabareunusuallycapable. Moreover,theyhavea commondesignand“feel”thatmakeiteasytolearnhowtousethem. Matlabissucha high-levellanguagethatprogramsareshort. Thismakesitpossibletoprovidecomplete programs in the text for all the examples. The programs are also provided in electronic formsothattheycanbeusedconvenientlyastemplatesforsimilarproblems. Inparticu- lar,thestudentisaskedtomodifysomeoftheseprogramsinexercises. Graphicsareapart ofthisPSE,sosolutionsaretypicallystudiedbyplottingthem. Matlabhassomesym- bolicalgebracapabilitiesbyvirtueofaMaplekernel(Maple1998). SolvingODEswith MATLAB exploits these capabilities in the analysis and solution of some of the examples andexercises. ThereisanInstructor’sManualwithsolutionsforalltheexercises. Most ofthesesolutionsinvolveaprogram,whichisavailabletoinstructorsinelectronicform. ThefirstODEsolverofMatlabwasbasedonaFORTRANprogramwrittenbyLarry ShampineandH.A.(Buddy)Watts. ForMatlab5,CleveMolerinitiatedalongandpro- ductiverelationshipbetweenShampineandTheMathWorks.Aresearchanddevelopment effort by Shampine and Mark Reichelt (1997) resulted in the Matlab ODE Suite. The ODE Suite has evolved considerably as a result of further work by Shampine, Reichelt, and Jacek Kierzenka (1999) and the evolution of Matlab itself. In particular, some of the IVP solvers were given the ability to solve differential algebraic equations (DAEs) of index1arising from singular mass matrices. Subsequently, Kierzenka and Shampine (2001)addedaprogramforsolvingBVPs. Mostrecently,SkipThompson,Shampine,and KierzenkaaddedaprogramforsolvingDDEswithconstantdelays(Shampine&Thomp- son 2001). We mention this history in part to express our gratitude to Cleve, Mark, and Jacek for the opportunity to work with them on software for this premier PSE and also tomakeclearthatwehaveauniqueunderstandingofthesoftwarethatunderliesSolving ODEswithMATLAB. EachofushasdecadesofexperiencesolvingODEsinbothacademicandnonacademic settings. Inthiswehavecontributedtothesubjectwellover200papersandhalfadozen books, but we have long wanted to write a book that makes our experience in advising peopleonhowtosolveODEsavailabletoawideraudience. SolvingODEswithMATLAB isthefulfillmentofthatwish. Weappreciatethehelpprovidedbymanyexpertswhohave commentedonportionsofthemanuscript. WayneEnrightandJacekKierzenkahavebeen especiallyhelpful.