Numerical Solution of Ordinary Differential Equations

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.