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The extended Pouzet-Runge-Kutta methods for nonlinear neutral delay-integro-differential equations. (English) Zbl 1203.65289

The authors investigate stability properties for numerical methods of the class noted in the title, for the equations

d dtyt-Nyt-τ=ft,yt,yt-τ, t-τ t gt,ξ,yξdξ,tt 0 ,

with a function y(t) given on the interval [t 0 -τ,t 0 ], in a d-dimensional complex space. These compound methods are created on the base of the classical Runge-Kutta (RK) methods for nonlinear ordinary differential equations, and a Pouzet quadrature formula for integrals in the right-hand side of the equation to be solved. The “nonlinear stability” of the extended Pouzet-Runge-Kutta methods means global and asymptotical stability in the Lyapunov sense. The obtained nonlinear stability results are based on the concept of “algebraic stability” of RK-methods by K. Burrage and J. C. Butcher [BIT, Nord. Tidskr. Inf.-behandl. 20, 185–203 (1980; Zbl 0431.65051 )]. Numerical examples illustrate the theoretical results.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
[1]Bellen A, Guglielmi N, Zennaro M (1999) On the contractivity and asymptotic stability of systems of delay differential equations of neutral type. Numer Math 39: 1–24 · Zbl 0917.65071 · doi:10.1023/A:1022361006452
[2]Brunner H, Vermiglio R (2003) Stability of solutions of delay functional integro-differential equations and their discretizations. Computing 71: 229–245 · Zbl 1049.65150 · doi:10.1007/s00607-003-0022-6
[3]Brunner H (2004) Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press, Cambridge
[4]Brunner H, van der Houwen PJ (1986) The numerical solution of Volterra equations. North-Holland, Amsterdam
[5]Burrage K, Butcher JC (1980) Nonlinear stability of a general class of differential equations methods. BIT 20: 185–203 · Zbl 0431.65051 · doi:10.1007/BF01933191
[6]Hairer E, Wanner G (1996) Solving ordinary differential equations II: stiff and differential-algebraic problems. Springer, Berlin
[7]Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge
[8]Huang C, Fu H, Li S, Chen G (1999) Stability analysis of Runge–Kutta methods for non-linear delay differential equations. BIT 39: 270–280 · Zbl 0930.65090 · doi:10.1023/A:1022341929651
[9]Jin J (2007) A numerical solution for neutral delay integro-differential equations with Volterra type. Math Appl 20(supplement):31–33
[10]Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations. Kluwer, Dordrecht
[11]Vermiglio R, Torelli L (2003) A stable numerical approach for implicit non-linear neutral delay differential equations. BIT 43: 195–215 · Zbl 1030.65078 · doi:10.1023/A:1023613425081
[12]Yuexin Y, Wen L, Li S (2007) Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations. Appl Math Comput 191: 543–549 · Zbl 1193.65123 · doi:10.1016/j.amc.2007.02.114
[13]Yuexin Y, Li S (2007) Stability analysis of Runge–Kutta methods for nonlinear neutral delay integro-differential equations. Sci China (Ser A) 50: 464–474 · Zbl 1126.65068 · doi:10.1007/s11425-007-2043-7
[14]Zennaro M (1997) Asymptotic stability analysis of Runge–Kutta methods for nonlinear systems of delay differential equations. Numer Math 77: 549–563 · Zbl 0886.65092 · doi:10.1007/s002110050300
[15]Zhang C, He Y (2009) The extended one-leg methods for nonlinear neutral delay-integro-differential equations. Appl Numer Math 59: 1409–1418 · Zbl 1163.65052 · doi:10.1016/j.apnum.2008.08.006
[16]Zhang C, Vandewalle S (2004) Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal 24: 193–214 · Zbl 1057.65104 · doi:10.1093/imanum/24.2.193
[17]Zhang C, Vandewalle S (2006) General linear methods for Volterra integro-differential equations with memory. SIAM J Sci Comput 27: 2010–2031 · Zbl 1104.65133 · doi:10.1137/040607058
[18]Zhang C, Qin T, Jin J (2009) An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations. Appl Math Comput 215: 548–556 · Zbl 1177.65197 · doi:10.1016/j.amc.2009.05.048