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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Part of Classics in Applied Mathematics

  • Date Published: December 1995
  • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • format: Paperback
  • isbn: 9780898713541

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  • This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.

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    Product details

    • Date Published: December 1995
    • format: Paperback
    • isbn: 9780898713541
    • length: 621 pages
    • dimensions: 255 x 178 x 32 mm
    • weight: 1.121kg
    • availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
  • Table of Contents

    List of Examples
    Preface
    1. Introduction. Boundary Value Problems for Ordinary Differential Equations
    Boundary Value Problems in Applications
    2. Review of Numerical Analysis and Mathematical Background. Errors in Computation
    Numerical Linear Algebra
    Nonlinear Equations
    Polynomial Interpolation
    Piecewise Polynomials, or Splines
    Numerical Quadrature
    Initial Value Ordinary Differential Equations
    Differential Operators and Their Discretizations
    3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results
    Green's Functions
    Stability of Initial Value Problems
    Conditioning of Boundary Value Problems
    4. Initial Value Methods. Introduction. Shooting
    Superposition and Reduced Superposition
    Multiple Shooting for Linear Problems
    Marching Techniques for Multiple Shooting
    The Riccati Method
    Nonlinear Problems
    5. Finite Difference Methods. Introduction
    Consistency, Stability, and Convergence
    Higher-Order One-Step Schemes
    Collocation Theory
    Acceleration Techniques
    Higher-Order ODEs
    Finite Element Methods
    6. Decoupling. Decomposition of Vectors
    Decoupling of the ODE
    Decoupling of One-Step Recursions
    Practical Aspects of Consistency
    Closure and Its Implications
    7. Solving Linear Equations. General Staircase Matrices and Condensation
    Algorithms for the Separated BC Case
    Stability for Block Methods
    Decomposition in the Nonseparated BC Case
    Solution in More General Cases
    8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method
    Reducing the Cost of the Newton Iteration
    Finding a Good Initial Guess
    Further Remarks on Discrete Nonlinear BVPS
    9. Mesh Selection. Introduction
    Direct Methods
    A Mesh Strategy for Collocation
    Transformation Methods
    General Considerations
    10. Singular Perturbations. Analytical Approaches
    Numerical Approaches
    Difference Methods
    Initial Value Methods
    11. Special Topics. Reformulation of Problems in 'Standard' Form
    Generalized ODEs and Differential Algebraic Equations
    Eigenvalue Problems
    BVPs with Singularities
    Infinite Intervals
    Path Following, Singular Points and Bifurcation
    Highly Oscillatory Solutions
    Functional Differential Equations
    Method of Lines for PDEs
    Multipoint Problems
    On Code Design and Comparison
    Appendix A. A Multiple Shooting Code
    Appendix B. A Collocation Code
    References
    Bibliography
    Index.

  • Authors

    Uri M. Ascher

    Robert M. M. Mattheij

    Robert D. Russell

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