Zhang, Chengjian A class of new Pouzet-Runge-Kutta-type methods for nonlinear functional integro-differential equations. (English) Zbl 1237.65146 Abstr. Appl. Anal. 2012, Article ID 642318, 21 p. (2012). Summary: This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given. Cited in 3 Documents MSC: 65R20 Numerical methods for integral equations 45D05 Volterra integral equations 45G05 Singular nonlinear integral equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations Software:RODAS × Cite Format Result Cite Review PDF Full Text: DOI References: [1] C. Zhang and S. Vandewalle, “Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations,” IMA Journal of Numerical Analysis, vol. 24, no. 2, pp. 193-214, 2004. · Zbl 1057.65104 · doi:10.1093/imanum/24.2.193 [2] C. Zhang and S. Vandewalle, “General linear methods for Volterra integro-differential equations with memory,” SIAM Journal on Scientific Computing, vol. 27, no. 6, pp. 2010-2031, 2006. · Zbl 1104.65133 · doi:10.1137/040607058 [3] Y. Yu, L. Wen, and S. Li, “Nonlinear stability of Runge-Kutta methods for neutral delay integro-differential equations,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 543-549, 2007. · Zbl 1193.65123 · doi:10.1016/j.amc.2007.02.114 [4] C. Zhang, T. Qin, and J. Jin, “An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 548-556, 2009. · Zbl 1177.65197 · doi:10.1016/j.amc.2009.05.048 [5] C. Zhang, T. Qin, and J. Jin, “The extended Pouzet-Runge-Kutta methods for nonlinear neutral delay-integro-differential equations,” Computing, vol. 90, no. 1-2, pp. 57-71, 2010. · Zbl 1203.65289 · doi:10.1007/s00607-010-0103-2 [6] H. Brunner and R. Vermiglio, “Stability of solutions of delay functional integro-differential equations and their discretizations,” Computing, vol. 71, no. 3, pp. 229-245, 2003. · Zbl 1049.65150 · doi:10.1007/s00607-003-0022-6 [7] H. Brunner, “The discretization of neutral functional integro-differential equations by collocation methods,” Zeitschrift für Analysis und ihre Anwendungen, vol. 18, no. 2, pp. 393-406, 1999. · Zbl 0951.65149 · doi:10.4171/ZAA/889 [8] H. Brunner, “High-order collocation methods for singular Volterra functional equations of neutral type,” Applied Numerical Mathematics, vol. 57, no. 5-7, pp. 533-548, 2007. · Zbl 1119.65121 · doi:10.1016/j.apnum.2006.07.006 [9] C. T. H. Baker, “A perspective on the numerical treatment of Volterra equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 217-249, 2000. · Zbl 0976.65121 · doi:10.1016/S0377-0427(00)00470-2 [10] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, vol. 15 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2004. · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 [11] H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland, Amsterdam, The Netherlands, 1986. · Zbl 0611.65092 [12] W. Wang, “A generalized Halanay inequality for stability of nonlinear neutral functional differential equations,” Jurnal of Inequalities and Applications, vol. 2010, Article ID 475019, 16 pages, 2010. · Zbl 1197.26040 · doi:10.1155/2010/475019 [13] L. Wen, Y. Yu, and W. Wang, “Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 347, no. 1, pp. 169-178, 2008. · Zbl 1161.34047 · doi:10.1016/j.jmaa.2008.05.007 [14] E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems, vol. 14 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2nd edition, 1996. · Zbl 0900.65238 [15] Dahlquist G. and Jeltsch R., “Generalized disks of contractivity for ex-plicit and implicit Runge-Kutta methods,” TRITA-NA Report 7906, The Royal Institute of Technology, Stockholm, Sweden, 1979. [16] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.