The (scalar) integro-differential equation of neutral type,
, , where is positive, nondecreasing, and weakly singular at (e.g. , ), is considered in the weighted Lebesgue space .
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.