New numerical approach for fractional differential equations

TL;DR

This paper introduces a corrected fractional Adams-Bashforth method that accurately accounts for the nonlinearity of kernels in various fractional differential equations, improving upon existing methods.

Contribution

It presents a mathematically correct version of the fractional Adams-Bashforth method that incorporates kernel nonlinearity for Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu cases.

Findings

The proposed method recovers standard cases when fractional order equals one.

It corrects inaccuracies in existing fractional Adams-Bashforth methods.

The new approach improves the mathematical consistency of fractional differential equation solutions.

Abstract

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario. The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power $\alpha=1$.