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

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