*(English)*Zbl 0744.65103

The (scalar) integro-differential equation of neutral type,

$t>0$, $x\left(\theta \right)=\phi \left(\theta \right)$ $(-r\le \theta <0)$, where $g$ is positive, nondecreasing, and weakly singular at $\theta =0$ (e.g. $g\left(\theta \right)={\left|\theta \right|}^{-p}$, $0<p<1$), is considered in the weighted Lebesgue space ${L}_{g}^{2}$.

Using approximation techniques introduced by *H. T. Banks* and *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.