Numerical Solution of Ordinary Differential Equations

ISBN-13: 9789401079075 / Angielski / Miękka / 2012 / 250 str.

L. Fox

Opis

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 some books are therefore needed which use quite simple mathematics but which nevertheless communicate the essence of the 'numerical sense' which is exhibited by the real computing experts and which is surely needed, at least to some extent, by all who use modern computers and modern numerical software. In that book we treated, at no great length, a variety of computational problems in which the material on ordinary differential equations occupied about 50 pages. At that time it was quite common to find books on numerical analysis, with a little on each topic ofthat field, whereas today we are more likely to see similarly-sized books on each major topic: for example on numerical linear algebra, numerical approximation, numerical solution ofordinary differential equations, numerical solution of partial differential equations, and so on. These are needed because our numerical education and software have improved and because our relevant problems exhibit more variety and more difficulty. Ordinary differential equa- tions are obvious candidates for such treatment, and the current book is written in this sense.

Spis treści

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.