Solving Ordinary Differential Equations I: Nonstiff Problems

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Springer Science & Business Media, Apr 16, 2008 - Mathematics - 528 pages

This book deals with methods for solving nonstiff ordinary differential equations. The first chapter describes the historical development of the classical theory, and the second chapter includes a modern treatment of Runge-Kutta and extrapolation methods. Chapter three begins with the classical theory of multistep methods, and concludes with the theory of general linear methods. The reader will benefit from many illustrations, a historical and didactic approach, and computer programs which help him/her learn to solve all kinds of ordinary differential equations. This new edition has been rewritten and new material has been included.

 

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Contents

Classical Mathematical Theory
1
I1 Terminology
2
I2 The Oldest Differential Equations
4
Leibniz and the Bernoulli Brothers
6
Variational Calculus
7
Clairaut
9
Exercises
10
I3 Elementary Integration Methods
12
Dense Output for the GBS Method
237
Control of the Interpolation Error
240
Exercises
241
II10 Numerical Comparisons
244
Performance of the Codes
249
A Stretched Error Estimator for DOP853
254
Effect of StepNumber Sequence in ODEX
256
II11 Parallel Methods
257

Second Order Equations
13
Exercises
14
I4 Linear Differential Equations
16
Variation of Constants
18
Exercises
19
I5 Equations with Weak Singularities
20
Nonlinear Equations
23
Exercises
24
I6 Systems of Equations
26
Fourier
29
Lagrangian Mechanics
30
Hamiltonian Mechanics
32
Exercises
34
I7 A General Existence Theorem
35
Existence Theorem of Peano
41
Exercises
43
I8 Existence Theory using Iteration Methods and Taylor Series
44
PicardLindelöf Iteration
45
Taylor Series
46
Recursive Computation of Taylor Coefficients
47
Exercises
49
19 Existence Theory for Systems of Equations
51
Vector Notation
52
Subordinate Matrix Norms
53
Exercises
55
I10 Differential Inequalities
56
The Fundamental Theorems
57
Estimates Using OneSided Lipschitz Conditions
60
Exercises
62
111 Systems of Linear Differential Equations
64
Resolvent and Wronskian
65
Inhomogeneous Linear Equations
66
Exercises
67
I12 Systems with Constant Coefficients
69
The Schur Decomposition
70
Numerical Computations
72
The Jordan Canonical Form
73
Geometric Representation
77
Exercises
78
I13 Stability
80
The RouthHurwitz Criterion
81
Computational Considerations
85
Liapunov Functions
86
Stability of Nonlinear Systems
87
Stability of NonAutonomous Systems
88
Exercises
89
I14 Derivatives with Respect to Parameters and Initial Values
92
The Derivative with Respect to a Parameter
93
Derivatives with Respect to Initial Values
95
The Nonlinear VariationofConstants Formula
96
Flows and VolumePreserving Flows
97
Canonical Equations and Symplectic Mappings
100
Exercises
104
I15 Boundary Value and Eigenvalue Problems
105
SturmLiouville Eigenvalue Problems
107
Exercises
110
I16 Periodic Solutions Limit Cycles Strange Attractors
111
Chemical Reactions
115
Limit Cycles in Higher Dimensions Hopf Bifurcation
117
Strange Attractors
120
The Ups and Downs of the Lorenz Model
123
Feigenbaum Cascades
124
Exercises
126
RungeKutta and Extrapolation Methods
129
II1 The First RungeKutta Methods
132
General Formulation of RungeKutta Methods
134
Discussion of Methods of Order 4
135
Optimal Formulas
139
Numerical Example
140
Exercises
141
II2 Order Conditions for RungeKutta Methods
143
The Derivatives of the True Solution
145
The Taylor Expansion of the True Solution
148
Faá di Brunos Formula
149
The Derivatives of the Numerical Solution
151
The Order Conditions
153
Exercises
154
II3 Error Estimation and Convergence for RK Methods
156
The Principal Error Term
158
Estimation of the Global Error
159
Exercises
163
II4 Practical Error Estimation and Step Size Selection
164
Embedded RungeKutta Formulas
165
Automatic Step Size Control
167
Starting Step Size
169
Numerical Experiments
170
Exercises
172
II5 Explicit RungeKutta Methods of Higher Order
173
6Stage 5th Order Processes
175
Embedded Formulas of Order 5
176
Higher Order Processes
179
Embedded Formulas of High Order
180
An 8th Order Embedded Method
181
Exercises
185
II6 Dense Output Discontinuities Derivatives
188
Continuous Dormand Prince Pairs
191
Dense Output for DOP853
194
Event Location
195
Discontinuous Equations
196
Numerical Computation of Derivatives with Respect to Initial Values and Parameters
200
Exercises
202
II7 Implicit RungeKutta Methods
204
Existence of a Numerical Solution
206
The Methods of Kuntzmann and Butcher of Order 2s
208
IRK Methods Based on Lobatto Quadrature
210
Collocation Methods
211
Exercises
214
II8 Asymptotic Expansion of the Global Error
216
Variable h
218
Negative h
219
Properties of the Adjoint Method
220
Symmetric Methods
221
Exercises
223
II9 Extrapolation Methods
224
The Aitken Neville Algorithm
226
The Gragg or GBS Method
228
Asymptotic Expansion for Odd Indices
231
Existence of Explicit RK Methods of Arbitrary Order
232
Order and Step Size Control
233
Parallel RungeKutta Methods
258
Parallel Iterated RungeKutta Methods
259
Extrapolation Methods
261
Exercises
263
II12 Composition of BSeries
264
BSeries
266
Order Conditions for RungeKutta Methods
269
Butchers Effective Order
270
Exercises
272
1113 Higher Derivative Methods
274
Collocation Methods
275
HermiteObreschkoff Methods
277
Fehlberg Methods
278
General Theory of Order Conditions
280
Exercises
281
1114 Numerical Methods for Second Order Differential Equations
283
Nyström Methods
284
The Derivatives of the Exact Solution
286
The Derivatives of the Numerical Solution
288
The Order Conditions
290
On the Construction of Nyström Methods
291
An Extrapolation Method for y¹¹fxy
294
Problems for Numerical Comparisons
296
Performance of the Codes
298
Exercises
300
II15 PSeries for Partitioned Differential Equations
302
Derivatives of the Exact Solution PTrees
303
PSeries
306
Order Conditions for Partitioned RungeKutta Methods
307
Further Applications of PSeries
308
Exercises
311
II16 Symplectic Integration Methods
312
Symplectic RungeKutta Methods
315
An Example from Galactic Dynamics
319
Partitioned RungeKutta Methods
326
Symplectic Nystrom Methods
330
Conservation of the Hamiltonian Backward Analysis
333
Exercises
337
II17 Delay Differential Equations
339
Constant Step Size Methods for Constant Delay
341
Variable Step Size Methods
342
Stability
343
An Example from Population Dynamics
345
Infectious Disease Modelling
347
An Example from Enzyme Kinetics
348
A Mathematical Model in Immunology
349
IntegroDifferential Equations
351
Exercises
352
Multistep Methods and General Linear Methods
354
III1 Classical Linear Multistep Formulas
356
Explicit Adams Methods
357
Implicit Adams Methods
359
Numerical Experiment
361
Explicit Nyström Methods
362
MilneSimpson Methods
363
Methods Based on Differentiation BDF
364
Exercises
366
III2 Local Error and Order Conditions
368
Order of a Multistep Method
370
Error Constant
372
Irreducible Methods
374
The Peano Kernel of a Multistep Method
375
Exercises
377
III3 Stability and the First Dahlquist Barrier
378
Stability of the BDFFormulas
380
Highest Attainable Order of Stable Multistep Methods
383
Exercises
387
III4 Convergence of Multistep Methods
391
Formulation as OneStep Method
393
Proof of Convergence
395
Exercises
396
III5 Variable Step Size Multistep Methods
397
Recurrence Relations for gⱼnϕⱼn and ϕⱼn
399
Variable Step Size BDF
400
General Variable Step Size Methods and Their Orders
401
Stability
402
Convergence
407
Exercises
409
III6 Nordsieck Methods
410
Equivalence with Multistep Methods
412
Implicit Adams Methods
417
BDFMethods
419
Exercises
420
III7 Implementation and Numerical Comparisons
421
Some Available Codes
423
Numerical Comparisons
427
III8 General Linear Methods
430
A General Integration Procedure
431
Stability and Order
436
Convergence
438
Order Conditions for General Linear Methods
441
Construction of General Linear Methods
443
Exercises
445
III9 Asymptotic Expansion of the Global Error
448
Asymptotic Expansion for Strictly Stable Methods 84
450
Weakly Stable Methods
454
The Adjoint Method
457
Symmetric Methods
459
Exercises
460
III10 Multistep Methods for Second Order Differential Equations
461
Explicit Störmer Methods
462
Implicit Störmer Methods
464
Numerical Example
465
General Formulation
467
Convergence
468
Asymptotic Formula for the Global Error
471
Rounding Errors
472
Exercises
473
Appendix Fortran Codes
475
Subroutine DOPRI5
477
Subroutine DOP853
481
Subroutine ODEX
482
Subroutine ODEX2
484
Driver for the Code RETARD
486
Subroutine RETARD
488
Bibliography
491
Symbol Index
521
Subject Index
523
Copyright

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Page 508 - Nova methodus pro maximis et minimis, itemque tangentibus, quae nee fractas nee irrationales quantitates moratur, et singulare pro illis calculi genus, die in den Actis Erudit.
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Page 493 - Analysis problematis antehac propositi, de inventione lineae descensus a corpore gravi percurrendae uniformiter, sic ut temporibus aequalibus aequales altitudines emetiatur: et alterius cujusdam Problematis Propositio, in: Acta eruditorum, Mai 1690, S.

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