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...

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