New numerical scheme of Atangana-Baleanu fractional integral: an application to groundwater flow within leaky aquifer

TL;DR

This paper introduces a novel numerical scheme for the Atangana-Baleanu fractional integral and applies it to model groundwater flow in a leaky aquifer, demonstrating its effectiveness in handling fractional derivatives.

Contribution

The paper presents the first numerical approximation for the Atangana-Baleanu fractional integral and applies it to a groundwater flow model with fractional derivatives.

Findings

The new scheme accurately approximates the fractional integral.

Application to groundwater flow shows the model's effectiveness.

Numerical results validate the scheme's precision.

Abstract

Many physical problems can be described using the integral equations known as Volterra equations. There exists quite a number of analytical methods that can handle these equations for linear cases. For non-linear cases, a numerical scheme is needed to obtain an approximation. Recently a new concept of fractional differentiation was introduced using the Mittag-Leffler function as kernel, and the associate fractional integral was also presented. Up to this point there is no numerical approximation of this new integral in the literature. Therefore, to accommodate researchers working in the field of numerical analysis, we propose in this paper a new numerical scheme for the new fractional integral. To test the accuracy of the new numerical scheme, we first revisit the groundwater model within a leaky aquifer by reverting the time classical derivative with the Atangana-Baleanu fractional derivative in Caputo sense. The new model is solved numerically by using this new scheme.