电子书简介、部分内容
This book has been developed from course notes that we wrote?? havingrepeatedly taught courses on the numerical solution of ordinary dierentialequations ODEs and related problems We have taught such courses at asenior undergraduate level as well as at the level of a rst graduate course onnumerical methods for dierential equations The audience typically consistsof students from Mathematics?? Computer Science and a variety of disciplinesin engineering and sciences such as Mechanical?? Electrical and Chemical Engineering?? Physics?? Earth Sciences?? etcThe material that this book covers can be viewed as a rst course on thenumerical solution of dierential equations It is designed for people whowant to gain a practical knowledge of the techniques used today The courseaims to achieve a thorough understanding of the issues and methods involvedand of the reasons for the successes and failures of existing software On onehand?? we avoid an extensive?? thorough?? theoremproof type exposition wetry to get to current methods?? issues and software as quickly as possibleOn the other hand?? this is not a quick recipe book?? as we feel that a deeperunderstanding than can usually be gained by a recipe course is required toenable the student or the researcher to use their knowledge to design theirown solution approach for any nonstandard problems they may encounter infuture work The book covers initialvalue and boundaryvalue problems?? aswell as dierentialalgebraic equations DAEs In a onesemester course wehave been typically covering over of the material it containsWe wrote this book partially as a result of frustration at not being ableto assign a textbook adequate for the material that we have found ourselvescovering There is certainly excellent?? indepth literature around In fact?? weare making repeated references to exhaustive texts which?? combined?? coveralmost all the material in this book Those books contain the proofs andreferences which we omit They span thousands of pages?? though?? and thetime commitment required to study them in adequate depth may be morethan many students and researchers can aord to invest We have tried tostay below a page limit and to address all three ODErelated areas meniiiiitioned above A signicant amount of additional material is covered in theExercises Other additional important topics are referred to in brief sectionsof Notes and References Software is an important and welldeveloped partof this subject We have attempted to cover the most fundamental softwareissues in the text Much of the excellent and publiclyavailable software isdescribed in the Software sections at the end of the relevant chapters?? andavailable codes are crossreferenced in the index Review material is highlighted and presented in the text when needed?? and it is also crossreferencedin the indexTraditionally?? numerical ODE texts have spent a great deal of time developing families of higher order methods?? eg RungeKutta and linear multistep methods?? applied rst to nonsti problems and then to sti problemsInitial value problems and boundary value problems have been treated inseparate texts?? although there is much in common There have been fundamental dierences in approach?? notation?? and even in basic denitions??between ODE initial value problems?? ODE boundary value problems?? andpartial dierential equations PDEsWe have chosen instead to focus on the classes of problems to be solved??mentioning wherever possible applications which can lend insight into therequirements and the potential sources of diculty for numerical solutionWe begin by outlining the relevant mathematical properties of each problemclass?? then carefully develop the lowerorder numerical methods and fundamental concepts for the numerical analysis Next we introduce the appropriate families of higherorder methods?? and nally we describe in some detailhow these methods are implemented in modern adaptive software An important feature of this book is that it gives an integrated treatment of ODEinitial value problems?? ODE boundary value problems?? and DAEs?? emphasizing not only the dierences between these types of problems but also thefundamental concepts?? numerical methods and analysis which they have incommon This approach is also closer to the typical presentation for PDEs??leading?? we hope?? to a more natural introduction to that important subjectKnowledge of signicant portions of the material in this book is essentialfor the rapidly emerging eld of numerical dynamical systems These are numerical methods employed in the study of the long term?? qualitative behaviorof various nonlinear ODE systems We have emphasized and developed inthis work relevant problems?? approaches and solutions But we avoided developing further methods which require deeper?? or more specic?? knowledgeof dynamical systems?? which we did not want to assume as a prerequisiteThe plan of the book is as follows Chapter is an introduction to thedierent types of mathematical models which are addressed in the bookWe use simple examples to introduce and illustrate initial and boundaryvalue problems for ODEs and DAEs We then introduce some importantapplications where such problems arise in practiceivEach of the three parts of the book which follow starts with a chapterwhich summarizes essential theoretical?? or analytical issues ie before applying any numerical method This is followed by chapters which developand analyze numerical techniques For initial value ODEs?? which compriseroughly half this book?? Chapter summarizes the theory most relevant forcomputer methods?? Chapter introduces all the basic concepts and simplemethods relevant also for boundary value problems and for DAEs?? Chapter is devoted to onestep RungeKutta methods and Chapter discussesmultistep methodsChapters are devoted to boundary value problems for ODEs Chapter discusses the theory which is essential to understand and to make eectiveuse of the numerical methods for these problems Chapter briey considers shootingtype methods and Chapter is devoted to nite dierenceapproximations and related techniquesThe remaining two chapters consider DAEs This subject has been researched and solidied only very recently in the past years Chapter is concerned with background material and theory It is much longer thanChapters and because understanding the relationship between ODEs andDAEs?? and the questions regarding reformulation of DAEs?? is essential andalready suggests a lot regarding computer approaches Chapter discussesnumerical methods for DAEsVarious courses can be taught using this book A week course can bebased on the rst chapters?? with an addition from either one of the remaining two parts In a week course or shorter in a more advanced graduateclass it is possible to cover comfortably Chapters and either Chapters or Chapters ?? with a more supercial coverage of the remaining materialThe exercises vary in scope and level of diculty We have provided somehints?? or at least warnings?? for those exercises that we or our students havefound more demandingMany people helped us with the tasks of shaping up?? correcting?? lteringand rening the material in this book First and foremost there are ourstudents in the various classes we taught on this subject They made usacutely aware of the dierence between writing with the desire to explainand writing with the desire to impress We note?? in particular?? G Lakatos??D Aruliah?? P Ziegler?? H Chin?? R Spiteri?? P Lin?? P Castillo?? E Johnson??D Clancey and D Rasmussen We have beneted particularly from ourearlier collaborations on other?? related books with K Brenan?? S Campbell??R Mattheij and R Russell Colleagues who have oered much insight?? adviceand criticism include E Biscaia?? G Bock?? C W Gear?? W Hayes?? C Lubich??V Murata?? D Pai?? J B Rosen?? L Shampine and A Stuart Larry Shampine??in particular?? did an incredibly extensive refereeing job and oered manycomments which have helped us to signicantly improve this text We havePREFACE valso beneted from comments of numerous anonymous referees
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