Numerical Methods for Delay Differential Equations

by Bellen, Alfredo; Zennaro, Marino

• ISBN13: 9780198506546
• ISBN10: 0198506546
• Format: Hardcover
• Copyright: 2003-05-29
• Publisher: Clarendon Press

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