Fractional order calculus or non-integer's order derivative is an old topic and three hundred years ago Leibniz was discussed for the first time with L’ Hospital. From the last two decades, fractional calculus got more consideration because of its physical meaning in different fields as physics, chemistry, electrical networks, biology, engineering, fluid flow, hydrology, and finance, etc. [1], [2], [3]. Many researchers have contributed very famous books, some of them are Miller and Ross [4], Oldham and Spanier [5], Podlubny [6], and Samko et al. [7]. The list of researchers such as Laplace, Able, Liouville, Riemann, Gr $\ddot{\mathrm{u}}$nwald, Letnikov, Levy and Riesz, etc, provided contribution and made a purely theoretical field of mathematics nowadays known as fractional calculus [8], [9], [10]. Many authors have worked in this field and on related questions. Many powerful techniques have been used to solve NL FDEs, some of them are (G’/G)-expansion [11], multiple exp-function method , protracted Fan sub-equation method [12], modified mathematical method [13], linear superposition method [14], time-spectrum function [15], auxiliary equation [16], kudryashov technique [17], functional variable method [18], [19], rational (G′/G)-expansion, improved $\left(G^{\prime }/G\right)$-expansion [20], improved modified extended tanh-function [21], simplest equation [22], modified simplest equation [23], local fractional decomposition [24], functional variable [25], extended $\left(G^{\prime }/G\right)$-expansion [26], transformed rational function [27], generalized $\exp \left(-\varphi \left(\xi \right)\right)-$expansion [28], modified Kudryashov [29], first integral approach [30], modified Jacobi elliptic expansion [31] generalized unified [32] and so on.

The solution of VO-FDEs are very important, especially in texture enhancement in medical imaging and analyzed the solutions by many investigators in recent years. The Akgül and Baleanu [33] researched the numerical reproducing kernel approach for the VO-FDEs. They proved their theory by providing the examples. Katsikadelis [34] established the numerical solution for VO-FDEs used both linear and NL types, and the VO fractional derivatives are in Caputo sense. They demonstrated that the approach is simple in implementation, effective and accurate. Bota and Căruntu [34] derived the new way of solution by polynomial least square method for nonlinear VO-FDEs and confirmed the accuracy based on serval examples. Yu et al. [35] investigated the VO fractional Centered difference scheme (VO-FCDS) by Riesz fractional order differential operator (RFDO) combine with Lagrange interpolation formula. The suggested method described the digital image processing in medical imaging. Tian and Liu [36] modified the well-known exp-function method applied to non-integers order Cohn-Allen equation successfully. They reduced the equation to system of algebraic equations and solved by Wu's method, obtained the solitary wave solution for the suggested equation. He et al. [37], [38] discussed the shape of the wave that affected by the unsmooth boundary conditions. The KdV and boussinesq equations are solved and obtained the solitary wave successfully. Also its shape is discussed by various fractal dimensions of the boundary. He et al. [39] presented the two-scale theory for the discontinuous problem. Also discussed the properties of two-scale theory. The fractional complex transform approach is used for the fractional differential equation to reduce it to system of differential equations which is discussed by He's and his co-authors [40], [41]. Han et al. [42] considered the KdV type problem with variable coefficients. They used the Fourier spectral method for KdV and modified KdV of non-integer order and presented results in the form of ball type, Kink type and periodic solitons solution. Tian and Liu [43] studied the fractional Fokas model and got the exact solution by the direct algebraic method. The topological soliton solutions are discussed for the differential equations based on the physical phenomena by Guner [44]. The applications of fractional complex transform and G’/G expansion method are very popular in the field of nonlinear fractional differential equations that using to find the exact traveling wave solutions [45], [46].

In this current study, finding the traveling wave solution for space-time VO-FDE. Such VO equations have a great importance in nonlinear sciences, and it is the most generalized form of FDE. This concept is completely new and have not been studied in the field of exact traveling solution. The VO space-time KdV equation has been considered to derive the closed form solutions in applied mathematics by modified $\left(G{}^{\prime }/G\right)-$expansion method to demonstrate the solution of the proposed VO-FDE. The nonlinear variable fractional order KdV equation has been resolved by many researchers with the help of different numerical, analytical techniques and other related studies to fractional order refer to [47], [48], [49].

The article is organized as in the following sections.

In section 2, the VO-CFD is discussed. In section 3, give the methodology of the modified (G’/G)-expansion method. Implementation of the suggested method to the fractional VO-KdV equation in section 3.1. In section 4, added the graphical image of the solutions and briefly explained the conclusion of the paper in the last section.

The VO integral and derivative operators are increasingly discussed and developed many definitions by different researchers [50], [51], [52]. Such as the He's fractional derivative, The two-scale fractional derivative, Riemann-Liouville fractional derivative [2], [53]. Here in this study, for the variable-order case we discuss the VO Caputo fractional derivative as:

And the properties are as follows:
$D_t^{\alpha \left(x,t\right)}f\left(t\right)=\frac{\Gamma \left(1-\gamma \right)}{\Gamma \left(1-\gamma +\alpha \left(x,t\right)\right)}{t}^{\gamma -\alpha \left(x,t\right)},$
$D_t^{\alpha \left(x,t\right)}\left(f\left(t\right)\mp h\left(t\right)\right)=D_t^{\alpha \left(x,t\right)}f\left(t\right)\mp D_t^{\alpha \left(x,t\right)}h\left(t\right)$
$D_t^{\alpha \left(x,t\right)}\left(f\left(t\right)h\left(t\right)\right)=h\left(t\right)D_t^{\alpha \left(x,t\right)}f\left(t\right)+f\left(t\right)D_t^{\alpha \left(x,t\right)}h\left(t\right),$

Consider the nonlinear VO- FDE of order $\alpha \left(x,t\right),\beta \left(x,t\right),\gamma \left(x,t\right)$is given by the form

where $Q$ is a polynomial in $u$, $D_t^{\alpha \left(x,t\right)},D_x^{\beta \left(x,t\right)},D_y^{\gamma \left(x,t\right)}$and $D^{\sigma \left(x,t\right)}$ represent VO fractional derivatives. The $(G^{\prime }/G)$-expansion method is following as [54]:

Taking the following fractional transformation

To converting VO-FDE into an ODE by the above transformation as follows:

Here, $k,l,m$ and $\omega$ are unknowns constants, prime represent the ODE in term of $\xi .$

Integrate Eq. (4) once or more and suppose that the soliton solution can be written as:

where $\psi \left(\xi \right)=\left(G^{\prime }/G\right),a_n\left(n=1,2,3,\dots ,M\right)$ can be zero, but $a_M$ cannot be zero, $\mu$ is an arbitrary constant. where $G=G\left(\xi \right)$ satisfy the below equation as:

$\psi \left(\xi \right)=h_1{G}^{-1}+h_{2}G,$
$\psi {\left(\xi \right)}^{\prime }=-h_1^2{G}^{-2}+h_2^{2}G,\text{etc}.$

Different types of solitary wave solution are generated for Eq. (6) [54], To find the values of $M$, we equate the highest order linear term with NL in (4). Then substituting (5) in (4) with (6) produce the equation of algebraic terms. Collecting the coefficients of same power of $G$ and obtained the system of algebraic equations. Simultaneously solving the algebraic equations to find these constants.