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Numerical methods for a class of singular integro-differential equations based on semigroup approximation. (English) Zbl 0744.65103
The (scalar) integro-differential equation of neutral type, $$(d/dt)(\int\sp 0\sb{-r} g(\theta)x(t+\theta)d\theta)=a\sb 0x(t)+\int\sp 0\sb{-r}a(\theta)x(t+\theta)d\theta+a\sb 1 x(t-r)+f(t),$$ $t>0$, $x(\theta)=\phi(\theta)$ $(-r\le \theta < 0)$, where $g$ is positive, nondecreasing, and weakly singular at $\theta=0$ (e.g. $g(\theta)=\vert \theta\vert\sp{-p}$, $0<p<1$), is considered in the weighted Lebesgue space $L\sp 2\sb g$. Using approximation techniques introduced by {\it H. T. Banks} and {\it J. A. Burns} [SIAM J. Control Optim., 18, 169-208 (1978; Zbl 0379.49025)], the convergence of the spline-based semi-discrete and fully- discrete numerical schemes is analyzed within the framework of semigroup theory (where the given integro-differential equation is formulated as a first-order hyperbolic partial differential equation with nonlocal boundary condition). Two examples (a singular neutral functional differential equation and an Abel-Volterra integral equation of the first kind) are employed to illustrate the feasibility of these numerical schemes.

65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
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