The (scalar) integro-differential equation of neutral type, \[ (d/dt)(\int^ 0_{-r} g(\theta)x(t+\theta)d\theta)=a_ 0x(t)+\int^ 0_{-r}a(\theta)x(t+\theta)d\theta+a_ 1 x(t-r)+f(t), \] \(t>0\), \(x(\theta)=\phi(\theta)\) \((-r\leq \theta < 0)\), where \(g\) is positive, nondecreasing, and weakly singular at \(\theta=0\) (e.g. \(g(\theta)=| \theta|^{-p}\), \(0<p<1\)), is considered in the weighted Lebesgue space \(L^ 2_ g\).
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.