An extrapolation method for a Volterra integral equation with weakly singular kernel

Convergence analysis of spectral collocation methods for a class of weakly singular Volterra integral equations

“…This type of equations arise in certain heat conduction problems with time dependent boundary conditions (see, e.g., [15]). Note that the kernel is singular at t ¼ 0 and s ¼ 0 if 0 < l < 1 and the kernel is singular at t ¼ 0 if l P 1.…”

“…An extrapolation algorithm based on the Euler method has been studied in [15]. In [7] high order product integration methods based on Newton-Cotes were proposed.…”

The exact solution of a class of Volterra integral equation with weakly singular kernel

“…(1) has been studied by several works for the case l > 1, e.g. product integration methods based on Newton-Cotes rules [3], Hermite-type collocation method [4], spline collocation and iterated collocation methods [5,6], extrapolation algorithm [7]. It's worth to note that for the case that m is a large positive integer (e.g.…”

Collocation Solutions of a Weakly Singular Volterra Integral Equation

“…For an extensive list of references on these and other approaches see [4]. Equation (1.3) has been the subject of several works for the case when µ > 1: certain classes of product integration methods based on Newton-Cotes rules were studied in [10]; Diogo et al [12] considered a fourth order Hermite-type collocation method and Lima and Diogo [17] developed an extrapolation algorithm, based on Euler's method. Recently, it was shown that general collocation methods on uniform meshes based on piecewise polynomials of degree m−1 yield convergence of order m. In [11] it was also proved that if certain derivatives of the exact solution are zero at the origin, then a higher order is attained at the mesh points by a special choice of the collocation points, like the Radau II points.…”