The modified simple equation method for solving some fractional-order nonlinear equations

Abstract

Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinear fractional Klein-Gordon equation by using the modified simple equation method.

A new method for solving nonlinear fractional differential equations

“…In literature, exact solutions of fractional differential equations have attracted the attention of researcher from different fields. Several research works have proposed techniques for solving fractional differential equations, such as G ′ /G−expansion method [3,28,30] , Exp−function method [12,29], first integral method [1,8,19,22], sub−equation method [13,24,32], Jacobi elliptic funtion method [9,11,26,31] , modified Kudryashov method [4,5,6,17,20], extended tanh method [7],modified simple equation method [18] and others.…”

A novel analytical method for time-fractional differential equations

“…For this reason, in recent years, several e¤ective mathematical methods have been proposed to obtain solutions of NFDEs. Such as the ( G 0 G )-expansion method [4,5,6], exp-rational function method [7], exp-function method [8,9], the modi…ed simple equation method [10], the generalized Kudryashov method [11], the …rst integral method [12], the simplest equation method [13], the functional variable method [14], sub-equation method [15], ansatz method [16,17], trial equation method [18] and so on [19]. The common of these analytical methods is based on the homogenous balance principle.…”

Regarding on the exact solutions for the nonlinear fractional differential equations

“…However some fresh solutions can be found with the usage of the MSE method, since φ is not a predefined function nor is it a solution of any pre-defined equation. This is the main advantage of the proposed method [22,23].…”