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An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations. (English) Zbl 1177.65197

The article deals with the Runge-Kutta method for the following neutral delay integro-differential equation

d dt[y(t)-Ny(t-τ)]=ft,y(t),y(t-τ)m t-τ t g(t,s,y(s))ds,tt 0 y(t)=φ(t),t 0 -τtt 0 ,

(φ:[t 0 -τ,t 0 ] d , f:[t 0 ,)× d × d × d , g:{(t,s):t[t 0 ,),s[t-τ,t]}× d d are given functions). More precisely, the authors study the asymptotic stability of the Runge-Kutta method under small perturbations of the initial function φ(t). The main result of the article desribes conditions under that the Runge-Kutta methods are globally and asymptotically stable. This result improves the analogous theorem by Y. Yu, L. Wen and S. Li [Appl. Math. Comput. 191, No. 2, 543–549 (2007)]. In the end of the article, a numerical example is considered.

MSC:
65R20Integral equations (numerical methods)
34K40Neutral functional-differential equations
References:
[1]Yu, Y.; Wen, L.; Li, S.: Nonlinear stability of Runge – Kutta methods for neutral delay integro-differential equations, Appl. math. Comput. 191, 543-549 (2007) · Zbl 1193.65123 · doi:10.1016/j.amc.2007.02.114
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[5]Zhang, C.; Vandewalle, S.: Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. comput. Appl. math. 164 – 165, 797-814 (2004) · Zbl 1047.65117 · doi:10.1016/j.cam.2003.09.013
[6]Zhang, C.; Vandewalle, S.: Stability analysis of Runge – Kutta methods for nonlinear Volterra delay-integro-differential equations, IMA J. Numer. anal. 24, 193-214 (2004) · Zbl 1057.65104 · doi:10.1093/imanum/24.2.193
[7]Zhang, C.; Vandewalle, S.: General linear methods for Volterra integro-differential equations with memory, SIAM J. Sci. comput. 27, 2010-2031 (2006) · Zbl 1104.65133 · doi:10.1137/040607058
[8]Yu, Y.; Li, S.: Stability analysis of Runge – Kutta methods for nonlinear neutral delay integro-differential equations, Sci. China (Ser. A) 50, 464-474 (2007) · Zbl 1126.65068 · doi:10.1007/s11425-007-2043-7
[9]Burrage, K.; Butcher, J. C.: Nonlinear stability of a general class of differential equations methods, Bit 20, 185-203 (1980) · Zbl 0431.65051 · doi:10.1007/BF01933191
[10]Zhang, C.: Nonlinear stability of natural Runge – Kutta methods for neutral delay differential equations, J. comput. Math. 20, 583-590 (2002) · Zbl 1018.65101
[11]Huang, C.; Fu, H.; Li, S.; Chen, G.: Stability analysis of Runge – Kutta methods for non-linear delay differential equations, Bit 39, 270-280 (1999) · Zbl 0930.65090 · doi:10.1023/A:1022341929651
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