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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

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By Uri M. Ascher, Robert M. M. Mattheij, and Robert D. Russell


1995 / xxvi + 595 pages / ISBN: 978-0-898713-54-1 / List Price $80.00 / SIAM Member Price $56.00 / Order Code CL13

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.

Graduate and senior undergraduate students in applied mathematics, computer science, and other areas in science and engineering, as well as researchers in mathematical methods and others who use mathematical models in areas such as earth sciences, physics, and mechanical engineering will benefit from reading this book.

This text is recommended for a graduate course on numerical methods for boundary value ordinary differential equations or for part of a course at the senior undergraduate level on the numerical solution of ordinary differential problems.

List of Examples; Preface; Chapter 1: Introduction. Boundary Value Problems for Ordinary Differential Equations; Boundary Value Problems in Applications; Chapter 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; Chapter 3: Theory of Ordinary Differential Equations. Existence and Uniqueness Results; Green's Functions; Stability of Initial Value Problems; Conditioning of Boundary Value Problems; Chapter 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; Chapter 5: Finite Difference Methods. Introduction; Consistency, Stability, and Convergence; Higher-Order One-Step Schemes; Collocation Theory; Acceleration Techniques; Higher-Order ODEs; Finite Element Methods; Chapter 6: Decoupling. Decomposition of Vectors; Decoupling of the ODE; Decoupling of One-Step Recursions; Practical Aspects of Consistency; Closure and Its Implications; Chapter 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; Chapter 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; Chapter 9: Mesh Selection. Introduction; Direct Methods; A Mesh Strategy for Collocation; Transformation Methods; General Considerations; Chapter 10: Singular Perturbations. Analytical Approaches; Numerical Approaches; Difference Methods; Initial Value Methods; Chapter 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.

ISBN: 9780898713541

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