Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

Abstract

In this paper, the fractional projective Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Burgers equation, the space-time fractional mKdV equation and time fractional biological population model. The solutions are expressed in terms of fractional hyperbolic functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The fractal index for the obtained results is equal to one. Counter examples to compute the fractal index are introduced in appendix. Keywords: fractional projective Riccati expansion method, nonlinear fractional differential equation, modified Riemann-Liouville derivative, exact solution, fractal index. Math Subject: 26A33, 34A08, 34K37, 35R11. This paper organized as follows: brief introduction of the fractional calculus and the description of the fractional projective Riccati expansion method are introduced in section 2. In section 3, the solution of the space-time fractional Burgers equation, the space-time fractional mKdV equation and time fractional biological population model are studied. In section 4, discussion and conclusion are presented. Preliminaries and Fractional projective Riccati expansion methodFractional Calculus is the generalizations of the classical calculus. It provides a redefinition of mathematical tools and it is very useful to deal with anomalous and frictional systems [35][36][37][38][39][40][41][42][43][44][45][46][47]. There are different kinds of fractional integration and differentiation operators. The most famous one is the Riemann-Liouville definition [1][2][3][4][5][6][7], which has been used in various fields of science and engineering successfully, but this definition leads to the result that constant function differentiation is not zero. Caputo put definitions which give zero value for fractional differentiation of constant function, but these definitions require that the function should be smooth and differentiable [1][2][3][4][5][6][7].Recently, Jumarie derived definitions for the fractional integral and derivative called modified Riemann-Liouville [30][31][32][33][34]. Some advantages can be cited to the modified

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

“…Also, we can obtain the same solutions (22) and (23) from the fractional Riccati expansion method [25]. When = 1, we obtain the classical combined KdV-mKdV equation…”

“…Many effective methods for obtaining numerical and analytical solutions of FDEs have been presented such as finite difference method [8,9], finite element method [10], Adomian decomposition method [11,12], differential transform method [13], variational iteration method [14][15][16], homotopy perturbation method [17,18], spectral methods [19,20], discontinuous Galerkin method [21], Kansa method [22], the fractional subequation method [23], and generalized fractional subequation method [24]. Abdel-Salam and Yousif [25] introduced the fractional Riccati expansion method to obtain analytical solutions of FDEs with constant coefficients. They solved the space-time fractional KdV equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation.…”

Approximate Solution to the Fractional Lane–Emden Type Equations

“…Based on the definition of the fractal index () x  which is usually determined in terms of gamma functions [42][43][44], Equations (4)-(6) could be modified to the following…”

“…There are many effective methods to treat numerical and analytical solutions of FDEs such as Adomian decomposition method, homotopy perturbation method, variational iteration method, finite difference method, Laplace transform, Fourier transform, generalized differential transform, Lie symmetry group, the fractional subequation method, the (G′/G)-expansion method, and the first integral method [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”

Approximate Solution to the Fractional Second-Type Lane-Emden Equation

“…So it is very important to find efficient methods for solving fractional differential equations. Finding analytical and approximate solutions of FDEs is one of the most useful approaches to understand the physical mechanism of natural phenomenon and dynamically processes modeled by FDEs (Abdel-Salam (2013), (2015a), (2015b). …”