Numerical Solution of Ordinary Differential Equations

Numerical Solution of Ordinary Differential Equations by L. Fox.

Nearly 20 years ago we produced a treatise (of about the same length as this book) entitled Computing methods for scientists and engineers. It was stated that most computation is performed by workers whose mathematical training stopped somewhere short of the 'professional' level, and that...

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Bibliographic Details
Main Author: Fox, L. (Author, http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht : Springer Netherlands : Imprint: Springer, 1987.
Edition:1st ed. 1987.
Series:Springer eBook Collection.
Subjects:
Differential equations.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 Introduction
  • 1.1 Differential equations and associated conditions
  • 1.2 Linear and non-linear differential equations
  • 1.3 Uniqueness of solutions
  • 1.4 Mathematical and numerical methods of solution
  • 1.5 Difference equations
  • 1.6 Additional notes
  • Exercises
  • 2 Sensitivity analysis: inherent instability
  • 2.1 Introduction
  • 2.2 A simple example of sensitivity analysis
  • 2.3 Variational equations
  • 2.4 Inherent instability of linear recurrence relations. Initial-value problems
  • 2.5 Inherent instability of linear differential equations. Initial-value problems
  • 2.6 Inherent instability: boundary-value problems
  • 2.7 Additional notes
  • Exercises
  • 3 Initial-value problems: one-step methods
  • 3.1 Introduction
  • 3.2 Three possible one-step methods (finite-difference methods)
  • 3.3 Error analysis: linear problems
  • 3.4 Error analysis and techniques for non-linear problems
  • 3.5 Induced instability: partial instability
  • 3.6 Systems of equations
  • 3.7 Improving the accuracy
  • 3.8 More accurate one-step methods
  • 3.9 Additional notes
  • Exercises
  • 4 Initial-value problems: multi-step methods
  • 4.1 Introduction
  • 4.2 Multi-step finite-difference formulae
  • 4.3 Convergence, consistency and zero stability
  • 4.4 Partial and other stabilities
  • 4.5 Predictor-corrector methods
  • 4.6 Error estimation and choice of interval
  • 4.7 Starting the computation
  • 4.8 Changing the interval
  • 4.9 Additional notes
  • Exercises
  • 5 Initial-value methods for boundary-value problems
  • 5.1 Introduction
  • 5.2 The shooting method: linear problems
  • 5.3 The shooting method: non-linear problems
  • 5.4 The shooting method: eigenvalue problems
  • 5.5 The shooting method: problems with unknown boundaries
  • 5.6 Induced instabilities of shooting methods
  • 5.7 Avoiding induced instabilities
  • 5.8 Invariant embedding for linear problems
  • 5.9 Additional notes
  • Exercises
  • 6 Global (finite-difference) methods for boundary-value problems
  • 6.1 Introduction
  • 6.2 Solving linear algebraic equations
  • 6.3 Linear differential equations of orders two and four
  • 6.4 Simultaneous linear differential equations of first order
  • 6.5 Convenience and accuracy of methods
  • 6.6 Improvement of accuracy
  • 6.7 Non-linear problems
  • 6.8 Continuation for non-linear problems
  • 6.9 Additional notes
  • Exercise
  • 7 Expansion methods
  • 7.1 Introduction
  • 7.2 Properties and computational importance of Chebyshev polynomials
  • 7.3 Chebyshev solution of ordinary differential equations
  • 7.4 Spline solution of boundary-value problems
  • 7.5 Additional notes
  • Exercises
  • 8 Algorithms
  • 8.1 Introduction
  • 8.2 Routines for initial-value problems
  • 8.3 Routines for boundary-value problems
  • 9 Further notes and bibliography
  • 10 Answers to selected exercises.

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