Stable and convergent difference schemes for weakly singular convolution integrals

We obtain new numerical schemes for weakly singular integrals of convolution type called Caputo fractional order using Taylor and series expansions grouping terms in a novel manner. A expansion argument is utilized to provide fractional-order approximations functions with minimal regularity. The resulting allow the approximation Cγ[0,T], where 0<γ≤5. mild invertibility criterion provided implicit schemes. Consistency stability are proven separately whole-number-order approximations. rate convergence time variable shown be O(τγ), 0<γ≤5 u∈Cγ[0,T], τ size partition mesh. Crucially, assumption integral kernel K being decreasing not required scheme converge second-order below Optimal results then both sets approximations, can up whole-number certain scenarios. Finally, examples that illustrate our findings.