Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

Abstract

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order.

An approximation method for fractional integro-differential equations

“…Recently many mathematicians and scientist have undertaken work on fractional calculus. One of these works, the Caputo-Fabrizo fractional derivative and its applications and simulations are given in the references [12][13][14][15]. The aim of this work here is to overcome the problems in approximating solutions of the following fractional order integro-differential equations with the boundary conditions, In order to solve this problem, the Sinc collocation method is used.…”

Atangana–Baleanu and Caputo Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second Grade Fluids over a Vertical Plate: A Comparative Study

“…Indeed, the claim of a singular kernel for the fractional derivative operator is not based on their observations; even they suggested their fractional derivative operator is appropriate for various physical problems. Itrat et al [12] employed the time-fractional Caputo-Fabrizio derivative on the advection-diffusion equation for tracing out the fundamental solutions using the Laplace transform. They investigated numerical solutions for the fractional diffusion phenomenon and normal advection-diffusion process.…”

Solution of fractional oxygen diffusion problem having without singular kernel

“…In this concern varies definitions of fractional derivative have been given till now. Recently the researchers described the new fractional derivative operator named Caputo-Fabrizio fractional derivative (see [3,4,7,12,17]). …”