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Zbl 1203.65289
Zhang, Chengjian; Qin, Tingting; Jin, Jie
The extended Pouzet-Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.
(English)
[J] Computing 90, No. 1-2, 57-71 (2010). ISSN 0010-485X; ISSN 1436-5057/e

The authors investigate stability properties for numerical methods of the class noted in the title, for the equations $$\frac{d}{dt}\left[y\left(t\right) - Ny\left( {t - \tau} \right)\right] = f\left(t,y\left( {t} \right),y\left( {t - \tau} \right),\int_{t - \tau} ^{t} {g\left( {t,\xi ,y\left( {\xi} \right)} \right)d\xi}\right),\;t \ge t_{0},$$ with a function $y(t)$ given on the interval $[t_{0} - \tau,t_{0}]$, 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 {\it K. Burrage} and {\it J. C. Butcher} [BIT, Nord. Tidskr. Inf.-behandl. 20, 185--203 (1980; Zbl 0431.65051 )]. Numerical examples illustrate the theoretical results.
MSC 2000:
*65R20 Integral equations (numerical methods)
45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations

Keywords: neutral delay-integro-differential equations; nonlinear stability; algebraic stability; Pouzet-Runge-Kutta methods; numerical examples

Citations: Zbl 0431.65051

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