  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 $\alpha=1$.
