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Book Categories |
Ch. I | Classical Mathematical Theory | |
I.1 | Terminology | 2 |
I.2 | The Oldest Differential Equations | 4 |
I.3 | Elementary Integration Methods | 12 |
I.4 | Linear Differential Equations | 16 |
I.5 | Equations with Weak Singularities | 20 |
I.6 | Systems of Equations | 26 |
I.7 | A General Existence Theorem | 35 |
I.8 | Existence Theory using Iteration Methods and Taylor Series | 44 |
I.9 | Existence Theory for Systems of Equations | 51 |
I.10 | Differential Inequalities | 56 |
I.11 | Systems of Linear Differential Equations | 64 |
I.12 | Systems with Constant Coefficients | 69 |
I.13 | Stability | 80 |
I.14 | Derivatives with Respect to Parameters and Initial Values | 92 |
I.15 | Boundary Value and Eigenvalue Problems | 105 |
I.16 | Periodic Solutions, Limit Cycles, Strange Attractors | 111 |
Ch. II | Runge-Kutta and Extrapolation Methods | |
II.1 | The First Runge-Kutta Methods | 132 |
II.2 | Order Conditions for Runge-Kutta Methods | 143 |
II.3 | Error Estimation and Convergence for RK Methods | 156 |
II.4 | Practical Error Estimation and Step Size Selection | 164 |
II.5 | Explicit Runge-Kutta Methods of Higher Order | 173 |
II.6 | Dense Output, Discontinuities, Derivatives | 188 |
II.7 | Implicit Runge-Kutta Methods | 204 |
II.8 | Asymptotic Expansion of the Global Error | 216 |
II.9 | Extrapolation Methods | 224 |
II.10 | Numerical Comparisons | 244 |
II.11 | Parallel Methods | 257 |
II.12 | Composition of B-Series | 264 |
II.13 | Higher Derivative Methods | 274 |
II.14 | Numerical Methods for Second Order Differential Equations | 283 |
II.15 | P-Series for Partitioned Differential Equations | 302 |
II.16 | Symplectic Integration Methods | 312 |
II.17 | Delay Differential Equations | 339 |
Ch. III | Multistep Methods and General Linear Methods | |
III.1 | Classical Linear Multistep Formulas | 356 |
III.2 | Local Error and Order Conditions | 368 |
III.3 | Stability and the First Dahlquist Barrier | 378 |
III.4 | Convergence of Multistep Methods | 391 |
III.5 | Variable Step Size Multistep Methods | 397 |
III.6 | Nordsieck Methods | 410 |
III.7 | Implementation and Numerical Comparisons | 421 |
III.8 | General Linear Methods | 430 |
III.9 | Asymptotic Expansion of the Global Error | 448 |
III.10 | Multistep Methods for Second Order Differential Equations | 461 |
Appendix. Fortran Codes | 475 | |
Bibliography | 491 | |
Symbol Index | 521 | |
Subject Index | 523 |
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