A nonstandard finite difference method for solving the fractional logistic model

This paper proposes an alternative solution formula for the logistic model, which is derived by substituting the exponential function with the Mittag-Leffler function in the solution of the first-order logistic model. Then, it developed two nonstandard finite difference approaches to solve the fractional logistic model. One method employed Mickens’s concepts to construct a nonstandard finite difference scheme, under the assumption that the analytical solution is unknown. The second method relies on the proposed analytical solution of the fractional logistic model. Surprisingly the two nonstandard finite difference algorithms are exactly the same. The convergence of the nonstandard finite difference scheme is proven by establishing its consistency and stability. Furthermore, it has been proven that the proposed numerical method is unconditionally stable. The performance of the method is demonstrated through two numerical examples selected from literature.

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Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).