The Numerical Integration of Neutral Functional-Differential Equations by Fully Implicit One-Step Methods

An algorithm for the numerical solution of neutral functional-differential equations is described. This algorithm is based on divided difference representation of fully implicit one-step methods. The resulting systems of nonlinear equations are solved using the predictor-corrector approach for nonstiff equations and by the modified Newton method for stiff equations. The step control strategy is based on local error estimation by comparing two approximations of successive orders. The details of implementations are described for systems of neutral delay-differential equations with state dependent delays, for Volterra integro-differential equations and for stiff delay-differential equations. The results of some numerical experiments on four test examples are presented and discussed. Ein Algorithmus zur numerischen Losung von neutralen Funktional-Differentialgleichungen wird beschrieben. Dieser Algorithmus basiert auf der dividierten Differenzendarstellung voll impliziter Einschrittverfahren. Die resultierenden Systeme nichtlinearer Gleichungen werden unter Verwendung der Pradiktor-Korrektor-Methode fur nichtsteife Gleichungen und durch das modifizierte Newtonverfahren fur steife Gleichungen gelost. Die Schrittweitenkontrollstrategie basiert auf lokaler Fehlerabschatzung durch Vergleich zweier Naherungen von aufeinanderfolgenden Ordnungen. Implementationsdetails werden fur Systeme neutraler Delay-Differentialgleichungen, fur Volterrasche lntegrodifferentialgleichungen und fur steife Delay-Differentialgleichungen beschrieben. Die Ergebnisse einiger numerischer Experimente fur vier Testbeispiele werden angegeben und diskutiert.

[1]  John C. Butcher.  The adaptation of STRIDE to delay differential equations , 1992 .

[2]  Z. Jackiewicz.  Quasilinear multistep methods and variable step predictor-corrector methods for neutral functional differential equations , 1986 .

[3]  R. Brayton.  Nonlinear oscillations in a distributed network , 1967 .

[4]  Zdzislaw Jackiewicz,et al.  The numerical solution of neutral functional differential equations by Adams predictor—corrector methods , 1991 .

[5]  Peter Linz,et al.  Linear Multistep Methods for Volterra Integro-Differential Equations , 1969, JACM.

[6]  Theodore A. Bickart.  P-stable andP[α, β]-stable integration/interpolation methods in the solution of retarded differential-difference equations , 1982 .

[7]  Marino Zennaro,et al.  P-stability properties of Runge-Kutta methods for delay differential equations , 1986 .

[8]  Marino Zennaro,et al.  Stability analysis of one-step methods for neutral delay-differential equations , 1988 .

[9]  S. Thompson,et al.  Software for the numerical solution of systems of functional differential equations with state-dependent delays , 1992 .

[10]  Alan Feldstein,et al.  Characterization of jump discontinuities for state dependent delay differential equations , 1976 .

[11]  Kenneth W. Neves.  Automatic Integration of Functional Differential Equations: An Approach , 1975, TOMS.

[12]  Christopher T. H. Baker,et al.  DELSOL: a numerical code for the solution of systems of delay-differential equations , 1992 .

[13]  Christopher T. H. Baker,et al.  The tracking of derivative discontinuities in systems of delay-differential equations , 1992 .

[14]  Christopher A. H. Paul,et al.  Developing a delay differential equation solver , 1992 .

[15]  A. Manitius,et al.  A fully-discrete spectral method for delay-differential equations , 1991 .

[16]  Alan Feldstein,et al.  High Order Methods for State-Dependent Delay Differential Equations with Nonsmooth Solutions , 1984 .

[17]  Z. Jackiewicz.  Fully implicit one-step methods for neutral functional-differential equations , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[18]  Zdzislaw Jackiewicz,et al.  ONE-STEP METHODS OF ANY ORDER FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS. , 1984 .