On the Fractional View Analysis of Keller–Segel Equations with Sensitivity Functions

Haobin Liu,Dumitru Baleanu,Rasool Shah,A A Alderremy,Hassan Khan,Shaban Aly

doi:10.1155/2020/2371019

Abstract

In this paper, the fractional view analysis of the Keller–Segal equations with sensitivity functions is presented. The Caputo operator has been used to pursue the present research work. The natural transform is combined with the homotopy perturbation method, and a new scheme for implementation is derived. The modified established method is named as the homotopy perturbation transform technique. The derived results are compared with the solution of the Laplace Adomian decomposition technique by using the systems of fractional Keller–Segal equations. The solution graphs and the table have shown that the obtained results coincide with the solution of the Laplace Adomian decomposition method. Fractional-order solutions are determined to confirm the reliability of the current method. It is observed that the solutions at various fractional orders are convergent to an integer-order solution of the problems. The suggested procedure is very attractive and straight forward and therefore can be modified to solve high nonlinear fractional partial differential equations and their systems.

Highlights

Fractional differential equations (FDEs) are the generalizations of the standard integer-order differential equations
FDEs have gained much attention in the recent decades, as they are broadly used in several areas to analyze different processes, such as the processing of the signal, control theory, identification of the system, fluid flow, biomathematics, and other fields [1,2,3]
E importance of FDEs is found in the literature because it can model most of the physical phenomena in science and engineering more accurately as compared to integer-order models [10,11,12], and the researchers have shown much interest to study fractional calculus and FDEs during the last decades

Summary

Introduction

Fractional differential equations (FDEs) are the generalizations of the standard integer-order differential equations. These techniques include the Laplace Adomian decomposition method (LADM) [16,17,18], Chebyshev wavelet methods (CWM) [19], collocation-shooting method [20], power series methods (PSM) [21], fractional Bernstein polynomials along with shooting method [22], fractional-Legendre spectral Galerkin method [23], variational iterative transform method (VITM) [24], homotopy perturbation transform method (HPTM) [25,26,27], homotopy analysis transform method (HATM) [28, 29], reduced differential transform method (RDTM) [30, 31], finite element technique (FET) [32], finite difference technique (FDT) [33], and q-homotopy analysis transform method (q-HATM) [34] Based on these techniques, a wide range of FDEs have been analyzed. In the current research paper, HPTM is implemented to solve fractional-order Keller–Segel equations. e solutions achieved through the suggested technique are straightforward and simple. e quality of the current method is appropriate to provide the analytical results to the given examples. e HPTM solutions are shown to be in close contact with the solutions of other existing techniques

Preliminary Concepts

Numerical Examples

Conclusion