Abstract
In this paper we consider implicit non-linear neutral delay differential equations to derive efficient numerical schemes with good stability properties. The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. Our hypothesis on the original equation allow us to study the boundedness and asymptotic stability of the true and numerical solutions by the theory of stability with respect to the forcing term.
Similar content being viewed by others
REFERENCES
U. M. Asher and L. R. Petzold, The numerical solution of delay-differentialalgebraic equations of retarded and neutral type, SIAM J. Numer. Anal., 32:5 (1995), pp. 1635-1657.
C. T. H. Baker, Retarded differential equations, J. CAM 125 1-2 (2000), pp. 309-335.
A. Bellen, N. Guglielmi, and M. Zennaro, On the contractivity and asymptotic stability of systems of delay differential equations of neutral type, BIT, 391 (1999), pp. 1-24.
A. Bellen and M Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
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.
H. Brunner and R. Vermiglio, Stability of solutions of neutral functional integrodifferential equations and their discretizations, Research Report, UDMI/RR/15/00, Univ. di Udine, Italy, 2000.
C. W. Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1971.
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
J. K. Hale and K. R. Meyer, A class of functional equations of neutral type, Mem. Amer. Math. Soc., 76 (1967).
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
J. K. Hale, E. F. Infante, and F. S. P Tsen, Stability in linear delay equations, J. Math. Anal. Appl., 105 (1985), pp. 533-555.
G. Da Hu and T. Mitsui, Stability analysis of numerical methods for systems of neutral delay-differential equations BIT, 35 (1995), pp. 504-515.
A. Iserles and J. Terjeki, Stability and asymptotic stability of functional-differential equations, J. London Math. Soc., 51:2 (1995), pp. 559-572.
V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, Dordrect, 1992.
L. M. Li, Stability of linear neutral delay-differential systems, Bull. Austral. Math. Soc. 38 (1988), pp. 339-344.
Y. Liu, Numerical solution of implicit neutral functional differential equations, SIAM J. Numer. Anal., 362 (1999), pp. 516-528.
W. Pinello and A. Ruehli, Time domain solutions for coupled problems using PEEC models with waveform relaxation, in Proc. IEEE Antennas and Propagation Society International Symposium 3, pp. 2118-2121, 1996.
R. Vermiglio and L. Torelli, Stability of non-linear neutral delay differential equations, Research Report, UDMI/RR/18/98, Univ. di Udine, 1998.
M. Zennaro, Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations, Numer. Math., 77 (1997), pp. 549-563.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vermiglio, R., Torelli, L. A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations. BIT Numerical Mathematics 43, 195–215 (2003). https://doi.org/10.1023/A:1023613425081
Issue Date:
DOI: https://doi.org/10.1023/A:1023613425081