Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC - Fractional Volterra integro-differential equations

This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana-Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana-Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana-Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight. (C) 2019 Elsevier Ltd. All rights reserved.