Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. (April 2020) - Hasan, Shatha; El-Ajou, Ahmad; Hadid, Samir; Al-Smadi, Mohammed; Momani, Shaher
Highlights: In this analysis, an attempt was made to study the fractional logistic model with the help of fractional derivative with Mittag-Leffler non-singular kernel, where the fractional derivative is considered in the ABC sense. The RKM has been applied to obtain approximate solutions for the non-linear fractional logistic deferential equation (FLDE). Three applications of this class of FLDEs are considered in the sense of ABC to verify the effectiveness of the presented method. Numerical and graphical results are also provided and conferred quantitatively to clarify the required solutions, where the results obtained are like those in previous studies that used Caputo type of fractional derivatives. So, since the ABC definition has a non-singular kernel, the use of ABC in DE modeling can be an appropriate substitute for Caputo fractional derivative and other fractional derivatives. We observed that the RKM method is very suitable, easy and effective to solve such a class of DEs and can be used to solve other types of differential equations. Abstract: In this article, a class of population growth model, the fractional nonlinear logistic system, is studied analytically and numerically. This model is investigated by means of Atangana-Baleanu fractional derivative with a non-local smooth kernel in Sobolev space. Existence and uniqueness theorem for the fractional logistic equation is provided based on the fixed-point theory. In this orientation, two numerical techniques are implemented to obtain the approximate solutions; the reproducing-kernel algorithm is based on the Schmidt orthogonalization process to construct a complete normal basis, while the successive substitution algorithm is based on an appropriate iterative scheme. Convergence analysis associated with the suggested approaches is provided to demonstrate the applicability theoretically. The impact of the fractional derivative on population growth is discussed by a class of nonlinear logistical models using the derivatives of Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Using specific examples, numerical simulations are presented in tables and graphs to show the effect of the fractional operator on the population curve as . The present results confirm the theoretical predictions and depict that the suggested schemes are highly convenient, quite effective and practically simplify computational time.