Piecewise mABC fractional derivative with an application

• Published in: AIMS mathematics Vol. 8; no. 10; pp. 24345 - 24366
• Main Authors: Khan, Hasib, Alzabut, Jehad, Gómez-Aguilar, J.F., Agarwal, Praveen
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2023
• Subjects: hyres-ulam stability; numerical simulations; picewise modified abc derivative; solution existence; tuberculosis
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.20231241

In this study, we give the notion of a piecewise modified Atangana-Baleanu-Caputo (mABC) fractional derivative and apply it to a tuberculosis model. This novel operator is a combination of classical derivative and the recently developed modified Atangana-Baleanu operator in the Caputo's sense. For this combination, we have considered the splitting of an interval $ [0, t_2] $ for $ t_2\in\mathbb{R}^+ $, such that, the classical derivative is applied in the first portion $ [0, t_1] $ while the second differential operator is applied in the interval $ [t_1, t_2] $. As a result, we obtained the piecewise mABC operator. Its corresponding integral is also given accordingly. This new operator is then applied to a tuberculosis model for the study of crossover behavior. The existence and stability of solutions are investigated for the nonlinear piecewise modified ABC tuberculosis model. A numerical scheme for the simulations is presented with the help of Lagrange's interpolation polynomial is then applied to the available data.