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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

$\left\{\begin{array}{c}\frac{d}{dt}\left[y\left(t\right)-Ny\left(t-\tau \right)\right]=f\left(t,y\left(t\right),y\left(t-\tau \right)m{\int }_{t-\tau }^{t}g\left(t,s,y\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds\right),\phantom{\rule{1.em}{0ex}}t\ge {t}_{0}\hfill \\ y\left(t\right)=\phi \left(t\right),\phantom{\rule{1.em}{0ex}}{t}_{0}-\tau \le t\le {t}_{0},\hfill \end{array}\right\$

($\phi :\phantom{\rule{4pt}{0ex}}\left[{t}_{0}-\tau ,{t}_{0}\right]\to {ℂ}^{d}$, $f:\phantom{\rule{4pt}{0ex}}\left[{t}_{0},\infty \right)×{ℂ}^{d}×{ℂ}^{d}×\to {ℂ}^{d}$, $g:\phantom{\rule{4pt}{0ex}}\left\{\left(t,s\right):\phantom{\rule{4pt}{0ex}}t\in \left[{t}_{0},\infty \right),s\in \left[t-\tau ,t\right]\right\}×{ℂ}^{d}\to {ℂ}^{d}$ are given functions). More precisely, the authors study the asymptotic stability of the Runge-Kutta method under small perturbations of the initial function $\phi \left(t\right)$. 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:
 65R20 Integral equations (numerical methods) 34K40 Neutral 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 [2] Kolmanovskii, V.; Myshkis, A.: Introduction to the theory and applications of functional differential equations, (1999) [3] Brunner, H.: Collocation methods for Volterra integral and related functional differential equations, (2004) [4] Brunner, H.; Vermiglio, R.: Stability of solutions of delay functional integro-differential equations and their discretizations, Computing 71, 229-245 (2003) · Zbl 1049.65150 · doi:10.1007/s00607-003-0022-6 [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 [12] Brunner, H.; Van Der Houwen, P. J.: The numerical solution of Volterra equations, (1986) [13] Hairer, E.; Wanner, G.: Solving ordinary differential equations II: Stiff and differential-algebraic problems, (1991)