Sharp Error Bounds for a Fractional Collocation Method for Weakly Singular Volterra Integral Equations with Variable Exponent

Abstract

Variable-exponent weakly singular Volterra integral equations of the second kind with integral kernels of the form \((t-s)^{-\alpha (t)}\) are considered. In Liang and Stynes (IMA J Numer Anal 19:drad072, 2023) it is shown that a typical solution of such an equation exhibits a weak singularity at the initial time \(t=0\), similarly to the case where \(\alpha (t)\) is constant. Our paper extends this analysis further by giving a decomposition for the exact solution. To solve the problem numerically, a fractional polynomial collocation method is applied on a graded mesh. The convergence of the collocation solution to the exact solution is analysed rigorously and it is proved that specific choices of the fractional polynomials and mesh grading yield optimal-order convergence of the computed solution. Superconvergence properties of the iterated collocation solution are also analysed. Numerical experiments illustrate the sharpness of our theoretical results.