×

zbMATH — the first resource for mathematics

The numerical integration of neutral functional-differential equations by fully implicit one-step methods. (English) Zbl 0830.65079
This paper deals with fully implicit one-step methods for the numerical solution of neutral functional-differential equations (NFDEs) and develops a divided difference representation of these formulas. The authors also discuss how to estimate the local discretization error in an efficient way by comparing two approximations of successive orders.
The predictor-corrector implementation of fully implicit one-step methods is described in a general context of NFDEs, with additional details of implementation for systems of neutral delay differential equations, for Volterra integro-differential equations, and for stiff delay differential equations. If the delay equation is stiff the system of the resulting nonlinear equations is solved by a variant of Newton’s method. Some numerical experiments are presented and discussed.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems
65L12 Finite difference and finite volume methods for ordinary differential equations
65R20 Numerical methods for integral equations
65L70 Error bounds for numerical methods for ordinary differential equations
34K05 General theory of functional-differential equations
34E13 Multiple scale methods for ordinary differential equations
45J05 Integro-ordinary differential equations
Software:
DELSOL
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz, Nat. Bureau of Standards, Appl. Math. Series 55, in: Handbook of mathematical functions (1964)
[2] Barwell, On the asymptotic behaviour of the solution of a differential-difference equation, Utilitas Math. 6 pp 189– (1974) · Zbl 0293.34096
[3] Bellen, Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math. 52 pp 605– (1988) · Zbl 0644.65049
[4] Bickart, P-stable and P[x, \(\beta\)]-stable integration/interpolation methods in the solution of retarded differential-difference equations, BIT 22 pp 464– (1982)
[5] Brayton, Nonlinear oscillations in a distributed network, Quart. Appl. Math. 24 pp 289– (1967) · Zbl 0166.35102
[6] Butcher, The adaptation of STRIDE to delay-differential equations, Appl. Numer. Math. 9 pp 415– (1992) · Zbl 0776.65049
[7] Feedstein, High order methods for state-dependent delay differential equations with nonsmooth solutions, SIAM J. Numer. Anal. 21 pp 844– (1984)
[8] Itǒ, A fully-discrete spectral method for delay-differential equations, SIAM J. Numer. Anal. 28 pp 1121– (1991)
[9] Jackiewicz, One-step methods of any order for neutral functional differential equations, SI A M J. Numer. Anal. 21 pp 486– (1984)
[10] Jackiewicz, Quasilinear multistep methods and variable step predictor-corrector methods for neutral functional differential equations, SIAM J. Numer. Anal. 23 pp 423– (1986) · Zbl 0602.65056
[11] Jackiewicz, One step methods for neutral delay-differential equations with state dependent delays, Zastos. Mat. 20 pp 445– (1990) · Zbl 0739.65065
[12] Jackiewicz , Z. Fully-implicit one-step methods for neutral functional-differential equations 1988 813 816
[13] Jackiewicz , Z. Lo , E. The numerical solution of neutral functional differential equations by Adams predictor-corrector methods 1988 · Zbl 0748.65057
[14] Jackiewicz , Z. Lo , E. The numerical integration of neutral functional-differential equations by fully implicit one-step methods 1991 · Zbl 0748.65057
[15] Kamont, On the Cauchy problem for differential-delay equations in a Banach space, Math. Nachr. 74 pp 173– (1976) · Zbl 0288.34069
[16] Linz, Linear multistep methods for Volterra integro-differential equations, J. Assoc. Comput. Machin. 16 pp 295– (1969) · Zbl 0183.45002 · doi:10.1145/321510.321521
[17] Neves, Automatic integration of functional differential equations: An Approach, ACM Trans. Math. Software 1 pp 357– (1975) · Zbl 0315.65045
[18] Neves, Characterization of jump discontinuities for state-dependent delay-differential equations, J. Math. Anal. Appl. 56 pp 689– (1976) · Zbl 0348.34054
[19] Neves, Software for the numerical solution of systems of functional differential equations with state-dependent delays, Appl. Numer. Math. 9 pp 385– (1992) · Zbl 0751.65045
[20] Paul, Developing a delay-differential equations solver, Appl. Numer. Math. 9 pp 403– (1992) · Zbl 0779.65043
[21] Shampine, The initial value problem (1975) · Zbl 0347.65001
[22] Smith, Periodic solutions of differential delay equations with threshold-type delays, Contemp. Math. 129 pp 153– (1992) · Zbl 0762.34044 · doi:10.1090/conm/129/1174140
[23] Willi, DELSOL-a numerical code for the solution of systems of delay-differential equations, Appl. Numer. Math. 9 pp 223– (1992)
[24] Willé, The tracking of derivative discontinuities in systems of delay-differential equations, Appl. Numer. Math. 9 pp 209– (1992)
[25] Zennaro, P-stability properties of Runge-Kutta methods for delay-differential equations, Numer. Math. 49 pp 305– (1986) · Zbl 0598.65056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.