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

Lane-Emden differential equations describe different physical and astrophysical phenomena that include forms of stellar structure, isothermal gas spheres, gas spherical cloud thermal history, and thermionic currents. This paper presents a computational approach to solve the problems related to fractional Lane-Emden differential equations based on neural networks. Such a solution will help solve the fractional polytropic gas spheres problems which have different applications in physics, astrophysics, engineering, and several real-life issues. We used Artificial Neural Network (ANN) framework in its feedforward back propagation learning scheme. The efficiency and accuracy of the presented algorithm are checked by testing it on four fractional Lane-Emden equations and compared with the exact solutions for the polytopic indices n=0,1,5 and those of the series expansions for the polytropic index n=3. The results we obtained prove that using the ANN method is feasible, accurate, and may outperform other methods.

Lane -Emden differential equation of the polytropic gas sphere could be used to construct simple models of stellar structures, star clusters and many configurations in astrophysics.This differential equation suffers from the singularity at the center and has an exact solution only for the polytropic index n=0, 1 and 5 . In the present paper, we present an analytical solution to the fractional polytropic gas sphere via accelerated series expansion. The solution is performed in the frame of conformable fractional derivatives. The calculated models recover the well-known series of solutions when 1 . Physical parameters such as mass-radius relation, density ratio, pressure ratio and temperature ratio for different fractional models have been calculated and investigated.We found that the present models of the conformable fractional stars have smaller volume and mass than that of both the integer star and fractional models performed in the frame of modified Rienmann Liouville derivatives.

Lane-Emden equation could be used to model stellar interiors, star clusters and many configurations in astrophysics. Unfortunately, there is an exact solution only for the polytropic index n=0, 1 and 5 . In the present paper, a series solution for the fractional Lane-Emden equation is presented. The solution is performed in the frame of modified Rienmann Liouville derivatives. The obtained results recover the well-known series solutions when 1 . Fractional model of n=3 has been calculated and mass-radius relation, density ratio, pressure ratio and temperature ratio have been investigated. We found that the fractional star has a smaller volume and mass than that of the integer star.

The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. 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.

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