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 . There are different kinds of fractional integration and differentiation operators. The most famous one is the Riemann-Liouville definition , 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 .Recently, Jumarie derived definitions for the fractional integral and derivative called modified Riemann-Liouville . Some advantages can be cited to the modified