×

zbMATH — the first resource for mathematics

Stability analysis of numerical methods for systems of neutral delay-differential equations. (English) Zbl 0841.65062
Authors’ abstract: Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicit \(A\)-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34K40 Neutral functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. V. Ahlfors,Complex Analysis, third edition, McGraw-Hill Co., New York, 1979. · Zbl 0395.30001
[2] C. T. H. Baker and C. A. H. Paul,Computing stability regions–Runge-Kutta methods for delay differential equations, IMA. J. Numer. Anal., 14 (1994), pp. 347–362. · Zbl 0805.65082 · doi:10.1093/imanum/14.3.347
[3] V. K. Barwell,Special stability problems for functional differential equations, BIT, 15 (1975), pp. 130–135. · Zbl 0306.65044 · doi:10.1007/BF01932685
[4] A. Bellen, Z. Jackiewicz and M. Zennaro,Stability analysis of one-step methods for neutral delay-differential equations, Numer. Math., 52 (1988), pp. 605–619. · Zbl 0644.65049 · doi:10.1007/BF01395814
[5] A. Bellen and M. Zennaro,Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math., 9 (1992), pp. 321–346. · Zbl 0749.65042 · doi:10.1016/0168-9274(92)90025-9
[6] R. K. Brayton and R. A. Willoughby,On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. Math. Anal. Appl., 18 (1967), pp. 182–189. · Zbl 0155.47302 · doi:10.1016/0022-247X(67)90191-6
[7] K. Gopalsamy,Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Boston, 1992. · Zbl 0752.34039
[8] J. K. Hale and S. M. Verduyn Lunel,Introduction to Functional Differential Equations, Springer Verlag, New York, 1993. · Zbl 0787.34002
[9] K. J. in’t Hout,The stability of \(\theta\)-methods for systems of delay differential equations, Ann. Numer. Math., 1 (1994), pp. 323–334. · Zbl 0837.65087
[10] K. J. in’t Hout and M. N. Spijker,Stability analysis of numerical methods for delay differential equations, Numer. Math., 59 (1991), pp. 807–814. · Zbl 0737.65073 · doi:10.1007/BF01385811
[11] Z. Jackiewicz,Asymptotic stability analysis of \(\theta\)-methods for functional equations, Numer. Math., 43 (1984), pp. 389–396. · Zbl 0557.65047 · doi:10.1007/BF01390181
[12] E. I. Jury,Inners and Stability of Dynamics Systems, John Wiley & Sons, New York, 1974. · Zbl 0307.93025
[13] T. Koto,A stability property of A-stable natural Runge-Kutta methods for systems of delay differential equations, BIT, 34 (1994), pp. 262–267. · Zbl 0805.65083 · doi:10.1007/BF01955873
[14] J. X. Kuang, J. X. Xiang and H. J. Tian,The asymptotic stability of one-parameter methods for neutral differential equations, BIT, 34 (1994), pp. 400–408. · Zbl 0814.65078 · doi:10.1007/BF01935649
[15] J. D. Lambert,Numerical Methods For Ordinary Differential Systems, John Wiley & Sons, New York, 1991. · Zbl 0745.65049
[16] M. Z. Liu and M. N. Spijker,The stability of the \(\theta\)-methods in the numerical solution of delay differential equations, IMA. J. Numer. Anal., 10 (1990), pp. 31–48. · Zbl 0693.65056 · doi:10.1093/imanum/10.1.31
[17] L. Torelli,Stability of numerical methods for delay differential equations, J. Comput. Appl. Math., 25 (1989), pp. 15–26. · Zbl 0664.65073 · doi:10.1016/0377-0427(89)90071-X
[18] M. Zennaro,P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49 (1986), pp. 305–318. · Zbl 0598.65056 · doi:10.1007/BF01389632
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.