Piecewise mABC fractional derivative with an application

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.