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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

$\frac{d}{dt}\left[y\left(t\right)-Ny\left(t-\tau \right)\right]=f\left(t,y\left(t\right),y\left(t-\tau \right),{\int }_{t-\tau }^{t}g\left(t,\xi ,y\left(\xi \right)\right)d\xi \right),\phantom{\rule{0.277778em}{0ex}}t\ge {t}_{0},$

with a function $y\left(t\right)$ given on the interval $\left[{t}_{0}-\tau ,{t}_{0}\right]$, 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.

MSC:
 65R20 Integral equations (numerical methods) 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations
RODAS
References:
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