Numerical Investigation of a Non-Singular Fractional Mathematical Model for HIV Infection of CD4$^{+}$ T Cells with Keller–Segel Chemotaxis

In this study, we introduce an efficient numerical approach for solving time-fractional differential equations that use the Atangana-Baleanu fractional derivative with the Mittag-Leffler kernel. We showcase the applicability of the proposed method via two representative time-fractional models: the HIV infection model that represents the dynamics of CD4+ T-cells and the Keller-Segel chemotaxis model. The first step of the procedure involves creating a numerical approximation for the Atangana-Baleanu fractional derivative and employing it to develop the Legendre operational matrix of fractional differential approximation. Once the proposed operational matrix is created, along with Legendre spectral collocation method, the governing fractional differential equation along with the associated initial and boundary conditions are converted into a system of nonlinear algebraic equations that we later solve to obtain solutions. Several numerical examples were presented in order to evaluate the effectiveness of the method developed and the obtained numerical results closely coincided with the correlated exact solutions that proved the correctness, efficiency, and proficiency of the proposed method of solving nonlinear time-fractional differential equations using the Mittag-Leffler kernel.