常微分方程和微分代數(shù)方程的計算機方法

前言

要使我國的數(shù)學事業(yè)更好地發(fā)展起來，需要數(shù)學家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學家創(chuàng)造更有利的發(fā)展數(shù)學事業(yè)的外部環(huán)境，這主要是加強對數(shù)學事業(yè)的支持與投資力度，使數(shù)學家有較好的工作與生活條件，其中也包括改善與加強數(shù)學的出版工作。從出版方面來講，除了較好較快地出版我們自己的成果外，引進國外的先進出版物無疑也是十分重要與必不可少的。從數(shù)學來說，施普林格（springer）出版社至今仍然是世界上最具權威的出版社?？茖W出版社影印一批他們出版的好的新書，使我國廣大數(shù)學家能以較低的價格購買，特別是在邊遠地區(qū)工作的數(shù)學家能普遍見到這些書，無疑是對推動我國數(shù)學的科研與教學十分有益的事。這次科學出版社購買了版權，一次影印了23本施普林格出版社出版的數(shù)學書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎數(shù)學書5本，應用數(shù)學書6本與計算數(shù)學書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學某方面的前沿，例如基礎數(shù)學中的數(shù)論、代數(shù)與拓撲三本，都是由該領域大數(shù)學家編著的“數(shù)學百科全書”的分冊。對從事這方面研究的數(shù)學家了解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數(shù)學類的書以“經(jīng)典”為主，應用和計算數(shù)學類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學家，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學獎”。這些大數(shù)學家的著作無疑將會對我國的科研人員起到非常好的指導作用。當然，23本書只能涵蓋數(shù)學的一部分，所以，這項工作還應該繼續(xù)做下去。更進一步，有些讀者面較廣的好書還應該翻譯成中文出版，使之有更大的讀者群。總之，我對科學出版社影印施普林格出版社的部分數(shù)學著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)容概要

Designed for those people who want to gain a practical knowledge of modem techniques，this book contains all the material necessary for a course on the nmnerical solution of differential equations.Written by two of the field's leading athorities，it provides a unified presentation of initial value and boundary value problems in ODEs as well as differential-algebraic equations.The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition.It also addresses reasons why existing software succeeds or fails.    This book is a practical and mathematically well informed introduction that emphasizes basic methods and theory，issues in the use and development of mathematical software，and examples from scientific engineering applications.Topics requiring an extensive amount of mathematical development，such as symplectic methods for Hamiltonian systems，are introduced，motivated，and included in the exercises，but a complete and rigorous mathematical presentation is referenced rather than included.    This book is appropriate for senior undergraduate or beginning graduate students with a computational focus and practicing engineers and scientists who want to learn about computational differential equations.A beginning course in numerical analysis is needed，and a beginning course in ordinary differential equations would be helpful.

作者簡介

作者：(美國)阿舍 (Uri M.Ascher) 譯者：(美國)Linda R.Petzold

書籍目錄

List of FiguresList of TablesPrefacePart Ⅰ：Introduction  1 Ordinary Differential Equations    1.1 IVPs    1.2 BVPs    1.3 Differential-Algebraic Equations    1.4 Families of Application Problems    1.5 Dynamical Systems    1.6 NotationPart Ⅱ：Initial Value Problems  2 On Problem Stability    2.1 Test Equation and General Definitions    2.2 Linear，Constant-Coefficient Systems    2.3 Linear，Variable-Coefficient Systems    2.4 Nonlinear Problems    2.5 Hamiltonian Systems    2.6 Notes and References    2.7 Exercises  3 Basic Methods，Basic Concepts    3.1 A Simple Method：Forward Euler    3.2 Convergence，Accuracy，Consistency，and O-Stability    3.3 Absolute Stability    3.4 Stiffness：Backward Euler      3.4.1 Backward Euler      3.4.2 Solving Nonlinear Equations    3.5 A-Stability，Stiff Decay    3.6 Symmetry：Trapezoidal Method    3.7 Rough Problems    3.8 Software，Notes，and References      3.8.1 Notes      3.8.2 Software    3.9 Exercises  4 One-Step Methods    4.1 The First Runge-Kutta Methods    4.2 General Formulation of Runge-Kutta Methods    4.3 Convergence，O-Stability，and Order for Runge-Kutta Methods    4.4 Regions of Absolute Stability for Explicit Runge-Kutta Methods    4.5 Error Estimation and Control    4.6 Sensitivity to Data Perturbations    4.7 Implicit Runge-Kutta and Collocation Methods      4.7.1 Implicit Runge-Kutta Methods Based on Collocation      4.7.2 Implementation and Diagonally Implicit Methods...      4.7.3 Order Reduction      4.7.4 More on Implementation and Singly Implicit RungeKutta Methods    4.8 Software，Notes，and References      4.8.1 Notes      4.8.2 Software    4.9 Exercises  5 Linear Multistep Methods    5.1 The Most Popular Methods      5.1.1 Adams Methods      5.1.2 BDF      5.1.3 Initial Values for Multistep Methods    5.2 Order，O-Stability，and Convergence      5.2.1 Order      5.2.2 Stability：Difference Equations and the Root Condition      5.2.3 O-Stability and Convergence    5.3 Absolute Stability    5.4 Implementation of hnplicit Linear Multistep Methods      5.4.1 Functional Iteration      5.4.2 Predictor-Corrector Methods      5.4.3 Modified Newton Iteration    5.5 Designing Multistep General-Purpose Software      5.5.1 Variable Step-Size Formulae      5.5.2 Estimating and Controlling the Local Error      5.5.3 Approximating the Solution at Off-Step Points    5.6 Software，Notes，and References      5.6.1 Notes      5.6.2 Software    5.7 ExercisesPart Ⅲ：Boundary Value Problems  6 More Boundary Value Problem Theory and Applications    6.1 Linear BVPs and Green's Function '.    6.2 Stability of BVPs    6.3 BVP Stiffness    6.4 Some Reformulation Tricks    6.5 Notes and References    6.6 Exercises  7 Shooting    7.1 Shooting：A Simple Method and Its Limitations      7.1.1 Difficulties    7.2 Multiple Shooting    7.3 Software，Notes，and References      7.3.1 Notes      7.3.2 Software    7.4 Exercises  8 Finite Difference Methods for Boundary Value Problems    8.1 Midpoint and Trapezoidal Methods      8.1.1 Solving Nonlinear Problems：Quasi-Linearization      8.1.2 Consistency，O-Stability，and Convergence    8.2 Solving the Linear Equations    8.3 Higher-Order Methods      8.3.1 Collocation      8.3.2 Acceleration Techniques    8.4 More on Solving Nonlinear Problems      8.4.1 Damped Newton      8.4.2 Shooting for Initial Guesses      8.4.3 Continuation    8.5 Error Estimation and Mesh Selection    8.6 Very Stiff Problems    8.7 Decoupling    8.8 Software，Notes，and References      8.8.1 Notes      8.8.2 Software    8.9 ExercisesPart Ⅳ：Differential-Algebraic Equations  9 More on Differential-Algebraic Equations    9.1 Index and Mathematical Structure      9.1.1 Special DAE Forms      9.1.2 DAE Stability    9.2 Index Reduction and Stabilization：ODE with Invariant      9.2.1 Reformulation of Higher-Index DAEs      9.2.2 ODEs with Invariants      9.2.3 State Space Formulation    9.3 Modeling with DAEs    9.4 Notes and References    9.5 Exercises  10 Numerical Methods for Differential-Algebraic Equations    10.1 Direct Discretization Methods      10.1.1 A Simple Method：Backward Euler      10.1.2 BDF and General Multistep Methods      10.1.3 Radau Collocation and Implicit Runge-Kutta Methods      10.1.4 Practical Difficulties      10.1.5 Specialized Runge-Kutta Methods for Hessenberg Index-2 DAEs    10.2 Methods for ODEs on Manifolds      10.2.1 Stabilization of the Discrete Dynamical System      10.2.2 Choosing the Stabilization Matrix F    10.3 Software，Notes，and References      10.3.1 Notes      10.3.2 Software    10.4 ExercisesBibliographyIndex

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