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Numerical stability of nonlinear delay differential equations of neutral type. (English) Zbl 0980.65077
The paper is devoted to the stability analysis of both the true solutions and the numerical approximations for systems of nonlinear neutral delay differential equations (NDDEs) of the form \(y'(t)=F(t,y(t),G(t,y(t-\tau (t)),y'(t-\tau (t))))\), \(t\geq t_0\), \(y(t)=g(t)\), \(t\leq t_0\), where \(F\) and \(G\) are complex continuous vector functions, \(g(t)\) is a \(C^1\)-continuous complex-valued function and \(\tau (t)\) is a continuous delay function such that \(\tau (t)\geq \tau_0 (t)>0\) and \(\alpha (t)=t-\tau (t)\) is increasing \(\forall t\geq t_0\). The main result is an extension of the results recently obtained by the authors [BIT 39, No. 1, 1-24 (1999; Zbl 0917.65071)] for the linear case. This is accomplished by considering a suitable reformulation of the given system, which transforms it into a nonlinear differential system coupled with an algebraic functional recursion. Numerical processes preserving the qualitative properties of the solutions are also investigated.
Reviewer: A.Dishliev (Sofia)

MSC:
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K40 Neutral functional-differential equations
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[1] Baker, C.T.H.; Paul, C.A.H.; Willè, D.R., Issues in the numerical solution of evolutionary delay differential equations, Adv. comput. math., 3, 171-196, (1995) · Zbl 0832.65064
[2] Bellen, A., Contractivity of continuous runge – kutta methods for delay differential equations, Appl. numer. math., 24, 219-232, (1997) · Zbl 0939.65100
[3] Bellen, A.; Guglielmi, N.; Zennaro, M., On the contractivity and asymptotic stability of systems of delay differential equations of neutral type, Bit, 39, 1-24, (1999) · Zbl 0917.65071
[4] Bellen, A.; Zennaro, M., Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. numer. math., 9, 321-346, (1992) · Zbl 0749.65042
[5] Brayton, R.K., Small signal stability criterion for networks containing lossless transmission lines, IBM J. res. develop., 12, 431-440, (1968) · Zbl 0172.20703
[6] Brayton, R.K.; Willoughby, R.A., On the numerical integration of a symmetric system of difference-differential equations of neutral type, J. math. anal. appl., 18, 182-189, (1967) · Zbl 0155.47302
[7] Guglielmi, N., Inexact Newton methods for the steady-state analysis of nonlinear circuits, Math. models methods appl. sci., 6, 43-57, (1996) · Zbl 0852.65065
[8] Hairer, E.; Zennaro, M., On error growth functions of runge – kutta methods, Appl. numer. math., 22, 205-216, (1996) · Zbl 0871.65071
[9] Hu, G.-Da; Mitsui, T., Stability analysis of numerical methods for systems of neutral delay-differential equations, Bit, 35, 504-515, (1995) · Zbl 0841.65062
[10] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press Boston
[11] Kuang, J.X.; Xiang, J.X.; Tian, H.J., The asymptotic stability of one-parameter methods for neutral differential equations, Bit, 34, 400-408, (1994) · Zbl 0814.65078
[12] Li, L.M., Stability of linear neutral delay-differential systems, Bull. austral. math. soc., 38, 339-344, (1988) · Zbl 0669.34074
[13] Pinello, W.; Cangellaris, A.C.; Ruehli, A., Hybrid electromagnetic modeling of noise interactions in packaged electronics based on the partial element equivalent circuit formulation, IEEE trans. microwave theory tech., 45, 1889-1896, (1997)
[14] A. Ruehli, U. Miekkala, A. Bellen, H. Heeb, Stable time domain solutions for EMC problems using PEEC circuit models, Proceedings of IEEE International Symposium on Electromagnetic Compatibility, Chicago, IL, 1994 pp. 371-376.
[15] Torelli, L., Stability of numerical methods for delay differential equations, J. comput. appl. math., 25, 15-26, (1989) · Zbl 0664.65073
[16] Torelli, L., A sufficient condition for GPN-stability for delay differential equations, Numer. math., 59, 311-320, (1991) · Zbl 0712.65079
[17] L. Torelli, R. Vermiglio, Stability of non-linear neutral delay differential equations, preprint, 1999. · Zbl 1030.65078
[18] Zennaro, M., Contractivity of runge – kutta methods with respect to forcing terms, Appl. numer. math., 10, 321-345, (1993) · Zbl 0774.65054
[19] M. Zennaro, Delay differential equations: theory and numerics, in: M. Ainsworth, W.A. Light, M. Marletta (Eds.), Theory and Numerics of Ordinary and Partial Differential Equations, Oxford University Press, Oxford, 1995 (Chapter 6). · Zbl 0847.34072
[20] Zennaro, M., Asymptotic stability analysis of runge – kutta methods for nonlinear systems of delay differential equations, Numer. math., 77, 549-563, (1997) · Zbl 0886.65092
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