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Wen, Liping; Wang, Suxia; Yu, Yuexin

Dissipativity of Runge-Kutta methods for neutral delay integro-differential equations. (English) Zbl 1177.65196

Appl. Math. Comput. 215, No. 2, 583-590 (2009).
The study of numerical methods for initial value problems of neutral delay integro-differential equations is more difficult and there are only a few papers dealing with such methods. This paper investigates numerical dissipativity of a class of nonlinear neutral delay integro-differential equations. Numerical dissipativity results for Runge-Kutta methods applied to an initial value problem of neutral delay integro-differential equation are established.
Applying an adaptation of an \(s\)-stage implicit Runge-Kutta method, using an interpolation procedure for the delay terms and the repeated trapezoidal rule for the integral terms, a new numerical scheme for solving the neutral delay integro-differential equations, are obtained. Under some hypothesis on the Runge-Kutta method (to be algebraically stable), the authors prove that the proposed numerical scheme for solving neutral delay integro-differential equations is dissipative (this means that the approximate values of the solution are bounded in norm). The obtained results are compared with existing results on the dissipativity for Runge-Kutta methods and for the dissipativity of \(\theta\)-methods.
Reviewer: Iulian Coroian (Baia Mare)

Cited in 16 Documents

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations

Keywords:

dissipativity; Runge-Kutta methods; algebraically stability; nonlinear neutral delay integro-differential equations
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\textit{L. Wen} et al., Appl. Math. Comput. 215, No. 2, 583--590 (2009; Zbl 1177.65196)
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References:

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