Stable and convergent difference schemes for weakly singular convolution integrals

Abstract

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order integrals using Taylor and fractional Taylor series expansions and grouping terms in a novel manner. A fractional Taylor series expansion argument is utilized to provide fractional-order approximations for functions with minimal regularity. The resulting schemes allow for the approximation of functions in $C^{\gamma }\left[0,T\right]$, where . A mild invertibility criterion is provided for the implicit schemes. Consistency and stability are proven separately for the whole-number-order approximations and the fractional-order approximations. The rate of convergence in the time variable is shown to be $O\left({\tau }^{\gamma }\right)$, for $u\in C^{\gamma }\left[0,T\right]$, where $\tau$ is the size of the partition of the time mesh. Crucially, the assumption of the integral kernel K being decreasing is not required for the scheme to converge in second-order and below approximations. Optimal convergence results are then proven for both sets of approximations, where fractional-order approximations can obtain up to whole-number rate of convergence in certain scenarios. Finally, numerical examples are provided that illustrate our findings.