A computationally efficient fractional predictor corrector approach involving the Mittag Leffler kernel
Sami Aljhani
TL;DR
This work addresses the numerical solution of fractional initial-value problems governed by the Atangana-Baleanu derivative with a Mittag-Leffler kernel. It introduces a computationally efficient predictor-corrector scheme based on Newton polynomial interpolation, utilizing an auxiliary midpoint and piecewise quadratic interpolation to construct the corrector alongside an explicit predictor. The authors derive closed-form integral weights and demonstrate superior accuracy and convergence on nonlinear AB-type problems compared to existing methods. Overall, the approach enriches the numerical toolkit for nonsingular, nonlocal fractional derivatives and is adaptable to other fractional derivative definitions.
Abstract
In this paper, based on Newton interpolation we have proposed a numerical scheme of predictor-corrector type in order to solve fractional differential equations with the fractional derivative involving the Mittag-Leffler function. We have added an auxiliary midpoint in each sub-interval, this allows us to use a piecewise quadratic Newton interpolation to derive the corrector scheme. The derivation of the schemes for the midpoint and the predictor is done by means of a piecewise linear Newton interpolation. We present some illustrative examples for initial value problems that involve fractional derivatives in the sense of Atangana-Baleanu. The results of numerical experiments show that the proposed scheme is a powerful technique to handle fractional differential equations with nonlinear terms that involve operators of Atangana-Baleanu type. Moreover, the proposed method significantly improves the numerical accuracy in comparison with other methods.