Author | Fox, L. author |
---|---|
Title | Numerical Solution of Ordinary Differential Equations [electronic resource] / by L. Fox, D. F. Mayers |
Imprint | Dordrecht : Springer Netherlands, 1987 |
Connect to | http://dx.doi.org/10.1007/978-94-009-3129-9 |
Descript | XII, 250 p. online resource |
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