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Tables of Contents for Solving Ordinary Differential Equations 1
Chapter/Section Title
Page #
Page Count
Classical Mathematical Theory
Terminology
2
2
The Oldest Differential Equations
4
8
Newton
4
2
Leibniz and the Bernoulli Brothers
6
1
Variational Calculus
7
2
Clairaut
9
1
Exercises
10
2
Elementary Integration Methods
12
4
First Order Equations
12
1
Second Order Equations
13
1
Exercises
14
2
Linear Differential Equations
16
4
Equations with Constant Coefficients
16
2
Variation of Constants
18
1
Exercises
19
1
Equations with Weak Singularities
20
6
Linear Equations
20
3
Nonlinear Equations
23
1
Exercises
24
2
Systems of Equations
26
9
The Vibrating String and Propagation of Sound
26
3
Fourier
29
1
Lagrangian Mechanics
30
2
Hamiltonian Mechanics
32
2
Exercises
34
1
A General Existence Theorem
35
9
Convergence of Euler's Method
35
6
Existence Theorem of Peano
41
2
Exercises
43
1
Existence Theory using Interation Methods and Taylor Series
44
7
Picard-Lindelof Iteration
45
1
Taylor Series
46
1
Recursive Computation of Taylor Coefficients
47
2
Exercises
49
2
Existence Theory for Systems of Equations
51
5
Vector Notation
52
1
Subordinate Matrix Norms
53
2
Exercises
55
1
Differential Inequalities
56
8
Introduction
56
1
The Fundamental Theorems
57
3
Estimates Using One-Sided Lipschitz Conditions
60
2
Exercises
62
2
Systems of Linear Differential Equations
64
5
Resolvent and Wronskian
65
1
Inhomogeneous Linear Equations
66
1
The Abel-Liouville-Jacobi-Ostrogradskii Identity
66
1
Exercises
67
2
Systems with Constant Coefficients
69
11
Linearization
69
1
Diagonalization
69
1
The Schur Decomposition
70
2
Numerical Computations
72
1
The Jordan Canonical Form
73
4
Geometric Representation
77
1
Exercises
78
2
Stability
80
12
Introduction
80
1
The Routh-Hurwitz Criterion
81
4
Computational Considerations
85
1
Liapunov Functions
86
1
Stability of Nonlinear Systems
87
1
Stability of Non-Autonomous Systems
88
1
Exercises
89
3
Derivatives with Respect to Parameters and Initial Values
92
13
The Derivative with Respect to a Parameter
93
2
Derivatives with Respect to Initial Values
95
1
The Nonlinear Variation-of-Constants Formula
96
1
Flows and Volume-Preserving Flows
97
3
Canonical Equations and Symplectic Mappings
100
4
Exercises
104
1
Boundary Value and Eigenvalue Problems
105
6
Boundary Value Problems
105
2
Sturm-Liouville Eigenvalue Problems
107
3
Exercises
110
1
Periodic Solutions, Limit Cycles, Strange Attractors
111
21
Van der Pol's Equation
111
4
Chemical Reactions
115
2
Limit Cycles in Higher Dimensions, Hopf Bifurcation
117
3
Strange Attractors
120
3
The Ups and Downs of the Lorenz Model
123
1
Feigenbaum Cascades
124
2
Exercises
126
6
Runge-Kutta and Extrapolation Methods
The First Runge-Kutta Methods
132
11
General Formulation of Runge-Kutta Methods
134
1
Discussion of Methods of Order 4
135
4
``Optimal'' Formulas
139
1
Numerical Example
140
1
Exercises
141
2
Order Conditions for Runge-Kutta Methods
143
13
The Derivatives of the True Solution
145
1
Conditions for Order 3
145
1
Trees and Elementary Differentials
145
3
The Taylor Expansion of the True Solution
148
1
Faa di Bruno's Formula
149
2
The Derivatives of the Numerical Solution
151
2
The Order Conditions
153
1
Exercises
154
2
Error Estimation and Convergence for RK Methods
156
8
Rigorous Error Bounds
156
2
The Principal Error Term
158
1
Estimation of the Global Error
159
4
Exercises
163
1
Practical Error Estimation and Step Size Selection
164
9
Richardson Extrapolation
164
1
Embedded Runge-Kutta Formulas
165
2
Automatic Step Size Control
167
2
Starting Step Size
169
1
Numerical Experiments
170
2
Exercises
172
1
Explicit Runge-Kutta Methods of Higher Order
173
15
The Butcher Barriers
173
2
6-Stage, 5th Order Processes
175
1
Embedded Formulas of Order 5
176
3
Higher Order Processes
179
1
Embedded Formulas of High Order
180
1
An 8th Order Embedded Method
181
4
Exercises
185
3
Dense Output, Discontinuties, Derivatives
188
16
Dense Output
188
3
Continuous Dormand & Prince Pairs
191
3
Dense Output for DOP853
194
1
Event Location
195
1
Discontinuous Equations
196
4
Numerical Computation of Derivatives with Respect to Initial Values and Parameters
200
2
Exercises
202
2
Implicit Runge-Kutta Methods
204
12
Existence of a Numerical Solution
206
2
The Methods of Kuntzmann and Butcher of Order 2s
208
2
IRK Methods Based on Lobatto Quadrature
210
1
Collocation Methods
211
3
Exercises
214
2
Asymptotic Expansion of the Global Error
216
8
The Global Error
216
2
Variable h
218
1
Negative h
219
1
Properties of the Adjoint Method
220
1
Symmetric Methods
221
2
Exercises
223
1
Extrapolation Methods
224
20
Definition of the Method
224
2
The Aitken - Neville Algorithm
226
2
The Gragg or GBS Method
228
3
Asymptotic Expansion for Odd Indices
231
1
Existence of Explicit RK Methods of Arbitrary Order
232
1
Order and Step Size Control
233
4
Dense Output for the GBS Method
237
3
Control of the Interpolation Error
240
1
Exercises
241
3
Numerical Comparisons
244
13
Problems
244
5
Performance of the Codes
249
5
A ``Stretched'' Error Estimator for DOP853
254
2
Effect of Step-Number Sequence in ODEX
256
1
Parallel Methods
257
7
Parallel Runge-Kutta Methods
258
1
Parallel Iterated Runge-Kutta Methods
259
2
Extrapolation Methods
261
1
Increasing Reliability
261
2
Exercises
263
1
Composition of B-Series
264
10
Composition of Runge-Kutta Methods
264
2
B-Series
266
3
Order Conditions for Runge-Kutta Methods
269
1
Butcher's ``Effective Order''
270
2
Exercises
272
2
Higher Derivative Methods
274
9
Collocation Methods
275
2
Hermite-Obreschkoff Methods
277
1
Fehlberg Methods
278
2
General Theory of Order Conditions
280
1
Exercises
281
2
Numerical Methods for Second Order Differential Equations
283
19
Nystrom Methods
284
2
The Derivatives of the Exact Solution
286
2
The Derivatives of the Numerical Solution
288
2
The Order Conditions
290
1
On the Construction of Nystrom Methods
291
3
An Extrapolation Method for y'' = f(x, y)
294
2
Problems for Numerical Comparisons
296
2
Performance of the Codes
298
2
Exercises
300
2
P-Series for Partitioned Differential Equations
302
10
Derivatives of the Exact Solution, P-Trees
303
3
P-Series
306
1
Order Conditions for Partitioned Runge-Kutta Methods
307
1
Further Applications of P-Series
308
3
Exercises
311
1
Symplectic Integration Methods
312
27
Symplectic Runge-Kutta Methods
315
4
An Example from Galactic Dynamics
319
7
Partitioned Runge-Kutta Methods
326
4
Symplectic Nystrom Methods
330
3
Conservation of the Hamiltonian; Backward Analysis
333
4
Exercises
337
2
Delay Differential Equations
339
17
Existence
339
2
Constant Step Size Methods for Constant Delay
341
1
Variable Step Size Methods
342
1
Stability
343
2
An Example from Population Dynamics
345
2
Infectious Disease Modelling
347
An Example from Enzyme Kinetics
248
101
A Mathematical Model in Immunology
349
2
Integro-Differential Equations
351
1
Exercises
352
4
Multistep Methods and General Linear Methods
Classical Linear Multistep Formulas
356
12
Explicit Adams Methods
357
2
Implicit Adams Methods
359
2
Numerical Experiment
361
1
Explicit Nystrom Methods
362
1
Milne-Simpson Methods
363
1
Methods Based on Differentiation (BDF)
364
2
Exercises
366
2
Local Error and Order Conditions
368
10
Local Error of a Multistep Method
368
2
Order of a Multistep Method
370
2
Error Constant
372
2
Irreducible Methods
374
1
The Peano Kernel of a Multistep Method
375
2
Exercises
377
1
Stability and the First Dahlquist Barrier
378
13
Stability of the BDF-Formulas
380
3
Highest Attainable Order of Stable Multistep Methods
383
4
Exercises
387
4
Convergence of Multistep Methods
391
6
Formulation as One-Step Method
393
2
Proof of Convergence
395
1
Exercises
396
1
Variable Step Size Multistep Methods
397
13
Variable Step Size Adams Methods
397
2
Recurrence Relations for gj(n), Φj(n) and Φ(n)
399
1
Variable Step Size BDF
400
1
General Variable Step Size Methods and Their Orders
401
1
Stability
402
5
Convergence
407
2
Exercises
409
1
Nordsieck Methods
410
11
Equivalence with Multistep Methods
412
5
Implicit Adams Methods
417
2
BDF-Methods
419
1
Exercises
420
1
Implementation and Numerical Comparisons
421
9
Step Size and Order Selection
421
2
Some Available Codes
423
4
Numerical Comparisons
427
3
General Linear Methods
430
18
A General Integration Procedure
431
5
Stability and Order
436
2
Convergence
438
3
Order Conditions for General Linear Methods
441
2
Construction of General Linear Methods
443
2
Exercises
445
3
Asymptotic Expansion of the Global Error
448
13
An Instructive Example
448
2
Asymptotic Expansion for Strictly Stable Methods (8.4)
450
4
Weakly Stable Methods
454
3
The Adjoint Method
457
2
Symmetric Methods
459
1
Exercises
460
1
Multistep Methods for Second Order Differential Equations
461
14
Explicit Stormer Methods
462
2
Implicit Stormer Methods
464
1
Numerical Example
465
2
General Formulation
467
1
Convergence
468
3
Asymptotic Formula for the Global Error
471
1
Rounding Errors
472
1
Exercises
473
2
Appendix. Fortran Codes
475
16
Driver for the Code DOPRI5
475
2
Subroutine DOPRI5
477
4
Subroutine DOP853
481
1
Subroutine ODEX
482
2
Subroutine ODEX2
484
2
Driver for the Code Retard
486
2
Subroutine Retard
488
3
Bibliography
491
30
Symbol Index
521
2
Subject Index
523