Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations
Tóm tắt
This paper deals with the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. Sufficient conditions are derived that guarantee the asymptotic stability of the exact solutions. Adaptations of classical Runge–Kutta and linear multistep methods are suggested for solving such systems with commensurate delays. Stability criteria are constructed for the asymptotic stability of these numerical methods and compared to the stability criteria derived for the continuous problem. It is found that, under suitable conditions, these two classes of numerical methods retain the stability of the continuous systems. Some numerical examples are given that illustrate the theoretical results.
Tài liệu tham khảo
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, Berlin (1993)
Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Kluwer, Dordrecht (1999)
in’t Hout, K.J.: Stability analysis of Runge–Kutta methods for systems of delay differential equations. IMA J. Numer. Anal. 17, 17–27 (1997)
Hu, G., Mitsui, T.: Stability analysis of numerical methods for systems of neutral delay-differential equations. BIT 35, 504–515 (1995)
Hu, G., Cahlon, B.: Estimations on numerically stable step-size for neutral delay differential systems with multiple delays. J. Comput. Appl. Math. 102, 221–234 (1999)
Tchangani, A.P., Dambrine, M., Richard, J.P.: Stability of linear differential equations with distributed delay. In: Proceedings of the 36th IEEE Conference on Decision and Control, California, 3779–3784, 1997
Qiu, L., Yang, B., Kuang, J.X.: The NGP-stability of Runge–Kutta methods for systems of neutral delay differential equations. Numer. Math. 81, 451–459 (1999)
Zhang, C., Zhou, S.: The asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay-differential equations. Sci. China Ser. A 41, 1153–1157 (1998)
Zhang, C., Zhou, S.: Stability analysis of LMMs for systems of neutral multidelay-differential equations. Comput. Math. Appl. 38, 113–117 (1999)
Tian, H., Kuang, J.: The asymptotic behaviour of theoretical solution for the differential equations with several delay terms. J. Shanghai Teachers Univ. 23, 1–10 (1994)
Baker, C.T.H., Ford, N.J.: Stability properties of a scheme for the approximate solution of a delay integro-differential equation. Appl. Numer. Math. 9, 357–370 (1992)
Koto, T.: Stability of Runge–Kutta methods for delay integro-differential equations. J. Comput. Appl. Math. 145, 483–492 (2002)
Huang, C., Vandewalle, S.: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distribulated delays. SIAM J. Sci. Comput. 25, 1608–1632 (2004)
Luzyanina, T., Engelborghs, K., Roose, D.: Computing stability of differential equations with bounded and distributed delays. Numer. Algorithms 34, 41–66 (2003)
Lancaster, P., Tismenetsky, M.: The Theory of Matrices. Academic, Orlando (1985)
Brunner, H., van der Houwen, P.: The Numerical Solution of Volterra Equations, CWI Monographs 3. North-Holland, Amsterdam (1986)
Baker, C.T.H., Ford, N.J.: Convergence of linear multistep methods for a class of delay integro-differential equations. In: Int. Series of Numerical Mathematics, Birkhauser, Basel vol. 86, pp. 47–59 (1988)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)