New numerical approach for fractional differential equations

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 α = 1. evolution of complex systems, Lévy statistics, fractional Brownian, fractional signal and image processing, electrode-electrolyte polarization, electromagnetic waves, filters motion, phase-locked loops and non-local phenomena have justified to give a better description of the phenomenon under investigation than models with the integer order derivative [1,10].Nowadays, there have been a lot studies on approximate methods for fractional differential equations. For instance, Dithelm et al. and Li et al., have reported some results on numerical fractional ordinary differential equations [7,12]. When seeking an approximate solution to the fractional order ordinary and partial differential equations, among many other choices that have been used are include the Adomian decomposition, homotopy perturbation and differential transform methods, for example, see [11,15]. A lot has been reported in the literature on various fractional derivatives, ranging from the Riemann-Liouville to Atangana-Baleanu fractional derivative versions [3,5,6]. Not only that, when numerically simulating such models, different numerical approximation techniques have been adopted in both space [16][17][18][19][20][21] and time [4,8,9].The Adams-Bashforth has been recognized as a great and powerful numerical method able to provide a numerical solution closer to the exact solution. This method was developed with the classical differentiation using the fundamental theorem of calculus and taking the difference between two times including t n+1 and t n . This method was later extended to the concept of fractional differentiation with Caputo and Riemann-Liouville derivatives, however, the adaptation was not mathematically correct as the kernel of fractional integration is nonlinear. In addition to this when the fractional order α = 1 with this fractional version, we do not recover the classical Adams-Bashforth numerical scheme. In this paper, we will propose a new Adams-Bashforth for fractional differentiation with Caputo, Caputo-Fabrizio and Atangana-Baleanu derivatives, this version takes into account the nonlinearity of the kernels including the power law for Riemann-Liouville case, the exponential decay law for Caputo-Fabrizio case and Mittag-Leffler for Atangana-Baleanu case. Indeed when the fractional order turns to 1 one is expe...

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations

“…Unlike what proposed in [2], where a shorter but sufficiently large memory is considered after an in-depth analysis, the authors of [1] replace the function f in each integral in the right-hand-side (rhs) of (1) by the linear interpolant polynomial on the nodes t n−1 and t n…”

“…It is also worthwhile to point out that the paper [1] is written in a such poor way which can cause further confusion to readers. For instance, it is necessary to observe that:…”

“…Nat. Phenom., 13(1), 2018], it is proposed a numerical method (see [1,Eq. (3.11)]) for solving fractional differential equations (FDEs) of (presumably) order 0 < α < 1…”

“…Obtaining approximations without having to compute a persistent memory term of increasing length is an ambitious goal in order to save computation and techniques for incorporating just a part of the memory term, known under the name of short or fixed memory principle, have been investigated by several authors (e.g., see [2,7,13]). However, the naive truncation of the memory term to a fixed interval leads to a severe loss of accuracy [7] and an appropriate selection of the length of the fixed memory (which must be sufficiently long) is necessary to keep the error under control; reducing the memory to very local information (just the last two grid-points are involved in the method proposed in [1]) unavoidably leads to catastrophic and unreasonable results.…”

Analysis of rubella disease model with non-local and non-singular fractional derivatives

“…The derivatives based on exponential appear naturally in many problems in nature as being able to describe the effect of fading memory. This class of derivative has been applied in several research papers for instance [5,7,13,15,16,[18][19][20]22]. However, it was noted by several experts in the field that, this new derivative does not have a non-local kernel as its corresponding integral is not fractional, thus a new kernel was suggested by Atangana and Baleanu [6] where after some manipulations, the exponential decay kernel was replaced by the generalized Mittag-Leffler kernel.…”