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Numerical Methods for Delay Differential Equations,9780198506546
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Numerical Methods for Delay Differential Equations

by ;
ISBN13:

9780198506546

ISBN10:
0198506546
Format:
Hardcover
Pub. Date:
5/29/2003
Publisher(s):
Oxford University Press, USA

Questions About This Book?

What version or edition is this?
This is the edition with a publication date of 5/29/2003.
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  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc.

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  • Numerical Methods for Delay Differential Equations
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Summary

The main purpose of the book is to introduce the readers to the numerical integration of the Cauchy problem for delay differential equations (DDEs). Peculiarities and differences that DDEs exhibit with respect to ordinary differential equations are preliminarily outlined by numerous examples illustrating some unexpected, and often surprising behaviors of the analytical and numerical solutions. The effect of various kinds of delays on the regularity of the solution is described and some essential existence and uniqueness results are reported. The book is centered on the various approaches existing in the literature, and develops an exhaustive error and well-posedness analysis for the general classes of one-step and multistep methods.

Table of Contents

Introduction
1(19)
DDEs versus ODEs: some examples
2(7)
Numerical solution of DDEs: is ODE theory enough?
9(11)
Order failure
11(1)
Stability failure
12(5)
A good method for DDEs
17(3)
Existence and regularity of solutions of DDEs
20(16)
Location of discontinuities and smoothing of the solution
21(11)
Primary and secondary discontinuities
21(3)
Vanishing and non-vanishing delays
24(1)
Bounded and unbounded delays
25(2)
Multiple delays
27(3)
State dependent delays
30(1)
Propagation of discontinuities in systems
30(2)
Existence and uniqueness of solutions
32(4)
A review of DDE methods
36(26)
The first approaches
37(3)
Preliminary results on continuous ODE methods
40(6)
The standard approach via continuous ODE methods
46(8)
DDEs of neutral type
49(2)
Other formulations of DDEs of neutral type
51(3)
Bellman's method of steps
54(2)
DDEs as abstract Cauchy problems
56(4)
Waveform relaxation methods
60(2)
The standard approach via continuous ODE methods
62(48)
DDEs with constant or non-vanishing time dependent delay
63(7)
DDEs with arbitrary time dependent delay
70(8)
DDEs with state dependent delay
78(12)
Tracking of discontinuities
89(1)
DDEs of neutral type
90(5)
NDDEs with constant or non-vanishing time dependent delay
91(2)
NDDEs with state dependent delay
93(2)
Other formulations of DDEs of neutral type
95(10)
Integration of NDDEs with free stepsize
100(5)
One-step versus multistep methods
105(2)
Some additional references and codes
107(3)
Continuous Runge--Kutta methods for ODEs
110(42)
Continuous extensions of RK methods
111(7)
Interpolants of the first class
118(16)
Collocation methods
121(2)
Natural continuous extensions
123(7)
An application of the NCEs
130(4)
Interpolants of the second class
134(11)
A first uniform correction procedure
135(5)
Explicit uniform corrections of NCEs
140(3)
Implicit uniform corrections of NCEs
143(2)
Some further references
145(1)
Direct construction of continuous RK methods
145(3)
An application to the local error estimate
148(4)
Runge--Kutta methods for DDEs
152(31)
The general delay case
153(3)
DDEs with constant delay. Connections with Bellman's method and superconvergence
156(2)
DDEs with non-vanishing time dependent delay. Constrained mesh method and superconvergence
158(6)
DDEs with vanishing time dependent delay
164(7)
The pantograph equation
165(6)
RK methods for NDDEs
171(12)
Other formulations of NDDEs
176(3)
The code RADAR5
179(4)
Local error estimate and variable stepsize
183(30)
Order of the advancing and error-estimating methods
184(4)
Error-estimating methods of order p1 = p -- 1
186(1)
Error-estimating methods of order p1 = p + 1
186(1)
Continuous error-estimating methods
187(1)
Error propagation and stepsize control
188(5)
The uniform order q = p
191(1)
The uniform order q = p -- 1
191(1)
Conclusions
192(1)
Implementation without overlapping
193(4)
The general delay case
197(7)
Stepsize control in the code RADAR5
204(9)
Stability analysis of Runge--Kutta methods for ODEs
213(35)
Linear error growth
213(2)
Some classical stability results for ODE methods
215(6)
A-stability
215(2)
AN-stability
217(1)
BN-stability
218(2)
Algebraic stability
220(1)
Asymptotic stability and contractivity of RK methods
221(8)
Constant and variable stepsize
222(1)
Error growth functions
222(4)
Superexponential functions
226(1)
Asymptotic stability
227(2)
Stability of interpolants of RK methods
229(4)
Characterization and existence of stable interpolants
230(3)
Stability concepts with respect to forcing terms
233(15)
The behavior of the test equations
234(3)
Stability definitions for RK methods
237(2)
Analysis of RK methods
239(4)
Further properties of stable RK methods
243(5)
Stability analysis of DDEs
248(42)
General non-linear DDEs
249(6)
Linear scalar test equations
255(11)
Description of the asymptotic stability region Sτ for real coefficients
258(1)
Description of the asymptotic stability region Sτ for complex coefficients
259(7)
Linear systems of DDEs
266(2)
General non-linear NDDEs
268(6)
Linear systems of NDDEs
274(6)
Linear scalar NDDEs
280(2)
Further classes of test equations
282(8)
DDEs with vanishing delay. The pantograph equation
283(2)
NDDEs with vanishing delay. The generalized pantograph equation
285(3)
Pure DDEs
288(2)
Stability analysis of Runge--Kutta methods for DDEs
290(84)
Linear error growth
291(3)
Generalizations of A-stability to DDEs
294(32)
P-stability
299(5)
P-contractivity
304(4)
GP-stability
308(4)
GP-contractivity
312(2)
FP-stability and FP-contractivity
314(7)
D-Stability
321(5)
Generalizations of A-stability to NDDEs
326(6)
NP-stability
328(3)
GNP-stability
331(1)
ND-stability
331(1)
Generalizations of AN-stability and BN-stability to DDEs
332(13)
PN-stability, GPN-stability and FPN-stability
335(4)
RN-stability, GRN-stability and FRN-stability
339(3)
Asymptotic stability via contractivity
342(3)
Boundedness and asymptotic stability for NDDEs
345(7)
Delay independent asymptotic stability for linear systems
352(3)
Delay dependent asymptotic stability for linear systems
355(3)
The pantograph equation
358(7)
Contractivity and boundedness
359(1)
Asymptotic stability
360(5)
Further stability investigations
365(9)
Using the Kreiss resolvent
365(2)
Pure DDEs
367(1)
Multiple delays
368(1)
Regularity
369(1)
Stability of the stepsize control mechanism
370(2)
Numerical approximation of the characteristic roots
372(2)
References 374(18)
Index 392


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