Mathematical modeling approach to the fractional Bergman's model

Abstract

This paper presents the solution for a fractional Bergman's minimal blood glucose-insulin model expressed by Atangana-Baleanu-Caputo fractional order derivative and fractional conformable derivative in Liouville-Caputo sense. Applying homotopy analysis method and Laplace transform with homotopy polynomial we obtain analytical approximate solutions for both derivatives. Finally, some numerical simulations are carried out for illustrating the results obtained. In addition, the calculations involved in the modified homotopy analysis transform method are simple and straightforward.

On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

“…Similar results were obtained in [30][31][32]. The use of differential equations of fractional order with derivatives β a D α and Cβ a D α in the modeling of biological processes (fractional analogue of the Bergman model), electrical circuits, motion of electrons under the action of the electric field (fractional analogue of the Drude model), as well as in the analysis of applied dynamic models (Rabinovich-Fabrikant attractor), is described in [33][34][35][36][37].…”

“…According to (36) we get W α (a) = 0 and, hence, according to the lemma, the solutions y 1 (t), y 2 (t), ..., y m (t) to Equation ( 33) are linearly independent. The theorem is proved.…”

$\alpha$-Differentiable functions in complex plane

“…His model has been applied for fractional harmonic oscillator problem, the fractional damped oscillator problem, and the forced oscillator problem in the one-dimensional fractional dynamics. Since the conformable derivative is theoretically very more comfortable to handle, in [28], the mathematical modeling method for the fractional Bergman's model which involves fractional conformable derivative in Liouville-Caputo sense, and the fractional operators of Attangana-Baleanu-Caputo fractional derivative, is introduced. In many problems, analytic and exact solutions of fractional differential equations are not available, and numerical solutions are possible.…”