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E-raamat: Solving Ordinary Differential Equations I: Nonstiff Problems

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Solving Ordinary Differential Equations I: Nonstiff ProblemsSuurem pilt
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This book deals with methods for solving nonstiff ordinary differential equations. The first chapter describes the historical development of the classical theory from Newton, Leibniz, Euler, and Hamilton to limit cycles and strange attractors. In a second chapter a modern treatment of Runge-Kutta and extrapolation methods is given. Also included are continuous methods for dense output, parallel Runge-Kutta methods, special methods for Hamiltonian systems, second order differential equations and delay equations. The third chapter begins with the classical theory of multistep methods, and concludes with the theory of general linear methods. Many applications from physics, chemistry, biology, and astronomy together with computer programs and numerical comparisons are presented. This new edition has been rewritten, errors have been eliminated and new material has been included. The book will be immensely useful to graduate students and researchers in numerical analysis and scientific computing, and to scientists in the fields mentioned above.

This book deals with methods for solving nonstiff ordinary differential equations. This new edition has been rewritten, errors have been eliminated and new material has been included. The book will be useful to graduate students and researchers in numerical analysis and scientific computing.

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From the reviews



"This is the revised version of the first edition of Vol. I published in 1987. .Vols. I and II (SSCM 14) of Solving Ordinary Differential Equations together are the standard text on numerical methods for ODEs. ...This book is well written and is together with Vol. II, the most comprehensive modern text on numerical integration methods for ODEs. It may serve a a text book for graduate courses, ...and also as a reference book for all those who have to solve ODE problems numerically." Zeitschrift für Angewandte Mathematik und Physik



" This book is a valuable tool for students of mathematics and specialists concerned with numerical analysis, mathematical physics, mechanics, system engineering, and the application of computers for design and planning" Optimization



" This book is highly recommended as a text for courses in numerical methods for ordinary differential equations and as a reference for the worker. It should be in every library, both academic and industrial." Mathematics and Computers

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


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