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

• Published in: Chaos, solitons and fractals Vol. 126; pp. 394 - 402
• Main Authors: Arqub, Omar Abu, Maayah, Banan
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.09.2019
• Subjects: Atangana–Baleanu fractional approach; Reproducing kernel algorithm; Volterra integro-differential equation; Atangana–Baleanu fractional approach; Volterra integro-differential equation; Reproducing kernel algorithm
• DOI: 10.1016/j.chaos.2019.07.023
• ISSN: 0960-0779

•In this analysis, by developed the reproducing kernel algorithm within the Atangana–Baleanu fractional operator, the numerical solutions of Volterra integro-differential equations are discussed with respect to initial conditions of necessity.•The solution methodology involves the use of couple Hilbert spaces for both range and domain space.•Numerical algorithm and procedure of solution are assembled compatibility with the optimal formulation of the problem.•The optimal profiles show the performance of the numerical solutions and the effect of the Atangana–Baleanu fractional operator in the obtained results.•In this approach, computational simulations are introduced to delineate the suitability, straightforwardness, and relevance of the calculations created. 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.