In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

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In this article, the new exact travelling wave solutions of the time-and space-fractional KdV-Burgers equation has been found. For this the fractional complex transformation have been implemented to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations, in the sense of the Jumarie's modified Riemann-Liouville derivative. Afterwards, the improved (G'/G)-expansion method can be implemented to celebrate the soliton solutions of KdV-Burger's equation of fractional order.

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In this article, we establish exact solutions for the BBM and the MBBM equations by using a generalized (G'/G )-expansion method. The generalized (G'/G )-expansion method was used to construct solitary wave and periodic wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the (G'/G )-expansion method is straightforward and powerful mathematical tool for solving nonlinear problems.

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The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case. 1. Introduction During recent years, fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1–3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials, polymers, and other areas of science are well described by fractional differential equation [4–7]. The most important advantage of using FDEs is their nonlocal property, which means that the next state of a system depends not only upon its current state but also upon all of its historical states [8, 9]. Recently, the fractional functional analysis has been investigated by many researchers [10, 11]. For example, the properties and theorems of Yang-Laplace transforms and Yang-Fourier transforms [12] and their applications to the fractional ordinary differential equations, fractional ordinary differential systems, and fractional partial differential equations have been discussed. Many powerful methods have been established and developed to obtain numerical and analytical solutions of FDEs, such as finite difference method [13], finite element method [14], Adomian decomposition method [15, 16], differential transform method [17], variational iteration method [18–20], homotopy perturbation method [21, 22], the fractional sub-equation method [23], and generalized fractional subequation method [24]. How to extend the existing methods to solve other FDEs is still an interesting and important research problem. Thanks to the efforts of many researchers, several FDEs have been investigated and solved, such as the impulsive fractional differential equations [25], space- and time-fractional advection-dispersion equation [26–28], fractional generalized Burgers’ fluid [29], and fractional heat- and wave-like equations [30], and so forth. The

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In this paper, we establish exact solutions for some nonlinear partial differential integral equations (PDIE). The (((G′)/G))-expansion method was used to construct travelling wave solutions of the generalized (1+1) dimensional and the generalized (2+1) dimensional Ito equations. In this method we take the advantage of general solutions of second order linear ordinary differential equation (LODE) to solve effectively many nonlinear evolution equations. The (((G′)/G))-expansion method presents a wider applicability for handling nonlinear wave equations.

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本文应用一种新的$(G'/G)$-展开法构建了非线性分数阶Klein-Gordon方程的更多、更一般的精确解.利用分数阶复变换,非线性分数阶Klein-Gordon方程被转化为非线性常微分方程.应用扩展的$(G'/G)$-展开法构建非线性分数阶Klein-Gordon方程精确解.得到了一系列新的显式解,包括双曲函数解,三角函数解和负幂次解,利用该方法获得了比以往更丰富的解. This paper discuss a new approach of $(G'/G)$-expansion method for constructing more general exact solutions of nonlinear fractional Klein-Gordon equation. By using the fractional complex transformation,the nonlinear fractional Klein-Gordon equation have been converted to nonlinear ordinary differential equation,we will apply the extended $(G'/G)$-expansion method to construct the exact solutions of nonlinear fractional Klein-Gordon equation,a series new explicit solutions were obtained,which include hyperbolic function,trigonometric and negative exponential solutions,more richer than before results

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(3+1)维时空分数阶偏微分方程mKdV-ZK方程精确解的构建重要而令人感兴趣.本文通过含三维空间、一维时间的分数阶复变换将分数阶mKdV-ZK 方程转化为非线性常微分方程,再引入新的辅助微分方程的解及其新的展开形式,构建了mKdV-ZK方程系列精确解. It is very important and interesting to construct the exact solutions of the (3+1)-dimensional space-time fractional mKdV-ZK equation.Firstly, by using fractional complex transformation, mKdV-ZK equation was transformed into a nonlinear ordinary deferential equation, and then introducing extends (G'/G)-expansion method to construct the exact solution. A series of new exact solution for mKdV-ZK equations have been obtained

首先，系统给出（G^{′}/G^{2}）-展开法、F-展开法、（exp）-展开法、改进的Kudryashov方法、直接截断法，构建偏微分方程的精确解的起源与研究现状的文献综述。接下来，采用对比方式给出上述五种广义的函数展开法在构建偏微分方程精确解的步骤。最后，通过上述五种广义的函数展开法中的（G^{′}/G^{2}）-展开法、（exp）-展开法构建(2 + 1)维Boiti-Leon-Pempinelli方程的精确解，并使用控制变量法进行数学实验分析了(2 + 1)维Boiti-Leon-Pempinelli方程中三个变量对于精确解的影响。
First, the system gives（G^{′}/G^{2}）-expansion method, F-expansion method, （exp）-expansion method, improved Kudryashov method, direct truncation method, to construct the literature review of the origin and research status of the exact solutions of partial differential equations. Next, the steps of constructing the exact solutions of the partial differential equations by the above five generalized function expansion methods are given in comparison. Finally, through the above five generalized （G^{′}/G^{2}）-expansion method, （exp）-expansion method in the function expansion method constructs the exact solution of the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation. The control variable method is used to analyze the influence of three variables on the exact solution in the (2 + 1)-dimensional Boiti-Leon-Pempinelli equation.

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In this paper, we show that the improved (G'/G)- expansion method is equivalent to the tanh method and gives the same exact solutions of nonlinear partial differential equations.

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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.