New exact solutions for some fractional order differential equations via improved sub-equation method

Berat Karaagac

doi:10.3934/dcdss.2019029

Abstract

In this paper, improved sub-equation method is proposed to obtain new exact analytical solutions for some nonlinear fractional differential equations by means of modified Riemann Liouville derivative. The method is applied to time-fractional biological population model and space-time fractional Fisher equation successfully. Finally, simulations of new exact analytical solutions are presented graphically.

Highlights

Fractional calculus, defined by the generalization of the order of classical calculus to the arbitrary real or complex order, is a concept which is as old and deep-rooted as classic calculus [5]
Towards the end of the 17th century, it has arised with some spec between L’Hospital and Leibnitz, developed with the studies of the renown mathematicians such as Laplace, Abel, Fourier, Liouville, Riemann, Grunwald and Letnikov [19]. In addition to these developments recently, Atangana and Baleanu suggested a new fractional order derivative, the new derivative based on the generalized MittagLeffler function and the fractional derivative has non-singular and nonlocal kernel [3]
Of the paper, we are going to present the application of improved sub-equation method to time-fractional biological model and space-time fractional Fisher equation and obtain results for these fractional differential equations

Summary

Introduction

Fractional calculus, defined by the generalization of the order of classical (traditional) calculus to the arbitrary real or complex order, is a concept which is as old and deep-rooted as classic calculus [5]. Towards the end of the 17th century, it has arised with some spec between L’Hospital and Leibnitz, developed with the studies of the renown mathematicians such as Laplace, Abel, Fourier, Liouville, Riemann, Grunwald and Letnikov [19]. In addition to these developments recently, Atangana and Baleanu suggested a new fractional order derivative, the new derivative based on the generalized MittagLeffler function and the fractional derivative has non-singular and nonlocal kernel [3]. Improved sub-equation method, modified Riemann Liouville derivative, exact analytical solution, time-fractional biological population model, space-time fractional Fisher equation

BERAT KARAAGAC

Conclusions