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常微分方程和微分代数方程的计算机方法
  • 书号:9787030234865
    作者:Uri M. Ascher
  • 外文书名:Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
  • 装帧:圆脊精装
    开本:B5
  • 页数:336
    字数:396
    语种:英文
  • 出版社:科学出版社
    出版时间:2016-05-09
  • 所属分类:O17 数学分析
  • 定价: ¥178.00元
    售价: ¥140.62元
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Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field’s leading authorities, 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.
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目录

  • List of Figures
    List of Tables
    Preface
    Part 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 Notation
    Part 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 Runge-Kutta 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 Implicit Linear Multistep Methods
    5.4.1 Functional Iteration
    5.4.2 Predictor-Corrector Methods
    5.4.3 Modified Newton Iteration
    5.5 Designing Multistep Ceneral-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 Exercises
    Part 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 Exercises
    Part 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 Exercises
    Bibliography
    Index
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