Numerical Solution of Boundary Value Problems for Ordinary Differential Equations
Published on 31. December 1995
Book
Paperback/Softback
620 pages
978-0-89871-354-1 (ISBN)
Description
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.
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.
More details
Series
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 228 mm
Width: 152 mm
Thickness: 32 mm
Weight
1120 gr
ISBN-13
978-0-89871-354-1 (9780898713541)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Content
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.
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.