Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system

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.

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