Piecewise mABC fractional derivative with an application

Hasib Khan; Jehad Alzabut; J.F. Gómez-Aguilar; Praveen Agarwal; Hasib Khan; Jehad Alzabut; J.F. Gómez-Aguilar; Praveen Agarwal

doi:10.3934/math.20231241

AIMS Mathematics

2023, Volume 8, Issue 10: 24345-24366. doi: 10.3934/math.20231241

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Research article

Piecewise mABC fractional derivative with an application

1.
Department of Mathematics and Sciences, Prince Sultan University, 11586, Riyadh, Saudi Arabia
2.
Department of Mathematics, Shaheed Benazir Bhutto Uniersity, Sheringal, Dir Upper, Khyber Pakhtunkhwa, Pakistan
3.
Department of Industrial Engineering, OSTİM Technical University, 06374 Ankara, Turkey
4.
CONACyT-Tecnológico Nacional de Mexico/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
5.
Department of Mathematics and International College of Engineering, Jaipur-303012, India
6.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE

Received: 03 May 2023 Revised: 26 July 2023 Accepted: 28 July 2023 Published: 15 August 2023
MSC : 34A08, 26A33, 34A34

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.
- picewise modified ABC derivative,
- tuberculosis,
- solution existence,
- Hyres-Ulam stability,
- numerical simulations
Citation: Hasib Khan, Jehad Alzabut, J.F. Gómez-Aguilar, Praveen Agarwal. Piecewise mABC fractional derivative with an application[J]. AIMS Mathematics, 2023, 8(10): 24345-24366. doi: 10.3934/math.20231241

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Abstract

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.

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