Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics Solutions to FNLEEs in mathematicalphysics

Purpose – Fractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional ð D α ξ G = G Þ -expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev – Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma – Tasso – Olver (STO) equation and the nonlinear space-time fractional Kolmogorov – Petrovskii – Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometricfunction, hyperbolicfunction and rationalfunction are claimed to befresh and furthergeneral in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering. Design/methodology/approach – The rational fractional ð D α ξ G = G Þ -expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U ð ξ Þ ¼ P ni ¼ 0 a i ð D α ξ G = G Þ i = P ni ¼ 0 b i ð D α ξ G = G Þ i . Findings – Achieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature. Originality/value – The rational fractional ð D α ξ G = G Þ -expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive. identifications,


Introduction
Fractional calculus originating from some speculations of Leibniz and L'Hospital in 1695 has gradually gained profound attention of many researchers for its extensive appearance in various fields of real world. Exact traveling wave solutions to fractional order nonlinear evolution equations (FNLEEs) are of fundamental and important in applied science because of their wide use to depict the nonlinear fractional phenomena and dynamical processes of nature world. The FNLEEs and their solutions in closed form play fundamental role in describing, modeling and predicting the underlying mechanisms related to the biology, bio-genetics, physics, solid state physics, condensed matter physics, plasma physics, optical fibers, meteorology, oceanic phenomena, chemistry, chemical kinematics, electromagnetic, electrical circuits, quantum mechanics, polymeric materials, neutron point kinetic model, control and vibration, image and signal processing, system identifications, the finance, acoustics and fluid dynamics [1][2][3]. The closed form wave solutions of these equations [4][5][6] are greatly helpful to realize the mechanisms of the complicated nonlinear physical phenomena as well as their further applications in practical life. Some attractive powerful approaches take into account in the recent research area related to fractional derivative associated problems [7][8][9]. Therefore, it has become the core aim in the research area of fractional related problems that how to develop a stable approach for investigating the solutions to FNLEEs in analytical or numerical form. Many researchers have offered different approaches to construct analytic and numerical solutions to FNLEEs as well as integer order and put them forward for searching traveling wave solutions, such as the He-Laplace method [10], the exponential decay law [11], the reproducing kernel method [12], the Jacobi elliptic function method [13], the À G 0 =G Á -expansion method and its various modifications [14][15][16][17][18], the exp-function method [19], the sub-equation method [20,21], the first integral method [22], the functional variable method [23], the modified trial equation method [24], the simplest equation method [25], the Lie group analysis method [26], the fractional characteristic method [27], the auxiliary equation method [28,29], the finite element method [30], the differential transform method [31], the Adomian decomposition method [32,33], the variational iteration method [34], the finite difference method [35], the homotopy perturbation method [36] and the He's variational principle [37], the new extended direct algebraic method [38,39], the Jacobi elliptic function expansion method [40], the conformable double Laplace transform [41] etc. But each method does not bear high acceptance for the lacking of productivity to construct the closed form solutions to all kind of FNLEEs. That is why; it is very much indispensable to establish new techniques.
In this study, we offer a newly established technique, called the rational fractional ðD α ξ G=GÞ-expansion method [42], to investigate closed form analytic wave solutions to some FNLEEs in the sense of conformable fractional derivative [43]. This effectual and reliable productive method shows its high performance through providing abundant fresh and general solutions to the suggested equations. The obtained solutions might bring up their importance through the contribution to analyze the inner mechanisms of physical complex phenomena of real world and make an acceptable record in the literature. According to this perception, Khalil has introduced α order fractional derivative of ψ as

Preliminaries and methodology
If the function ψ is α differentiable in ð0; rÞ for r > 0 and lim x→0 þ T α ψðxÞ exists, then the conformable derivative at x ¼ 0 is defined as T α ψð0Þ ¼ lim x→0 þ T α ψðxÞ. The conformable integral of ψ is This integral represents usual Riemann improper integral.

Methodology
In this subsection, we discuss the main steps of the rational fractional ðD α ξ G=GÞ-expansion method to examine exact traveling wave solutions to FNLEEs. A fractional partial differential equation in the independent variables t; x 1 ; x 2 ; . . . ; x n is supposed to be as follows: . . . ; x n Þ, i ¼ 1; 2; 3; . . . ; k are unknown functions, F is a polynomial in u i and it's various partial derivatives of fractional order. Maintain the following steps to unravel Eqn (2.2.1) by the rational fractional ðD α ξ G=GÞ-expansion technique. Let us consider the nonlinear fractional composite transformation which reduces Eqn (2.2.1) to the following ordinary differential equation of fractional order with respect to the variable ξ: We might take anti-derivative of Eqn (2.2.3) term by term as many times as possible and integral constant can be set to zero as soliton solutions are sought.
Step 1: Suppose the traveling wave solution of Eqn (2.2.1) can be expressed as follows: where a 0 i s and b i ; s are unknown constants to be determined later and G ¼ GðξÞ satisfies the following auxiliary nonlinear ordinary differential equation of fractional order: ξ GðξÞ þ λD α ξ GðξÞ þ μGðξÞ ¼ 0; ( where C 1 and C 2 are arbitrary constants. Step 2: The positive constant n can be determined by taking homogenous balance between the highest order linear and nonlinear terms appearing in Eqn (2.2.3).
Step 3: Substitute (2.2.4) and (2.2.5) into Eqn (2.2.3) with the value of n obtained in step 2, we obtain a polynomial in ðD α ξ G=GÞ. Setting each coefficient of the resulted polynomial to zero gives a set of algebraic equations for a 0 i s and b i ; s by means of the symbolic computation software, such as Maple, provides the values of constants.
Step 4: Inserting the values of a

The nonlinear space-time fractional PKP equation
This well-known equation is given as With the aid of the fractional compound transformation uðx; y; tÞ Eqn (3.1.1) is turned into the following ordinary differential equations of fractional order due to the variable ξ: Considering the homogenous balance to Eqn (3.1.4), the solution (2.2.4) becomes Eqn (3.1.4) together with (3.1.5) and (2.2.5) becomes a polynomial in ðD α ξ G=GÞ equating whose coefficients to zero and solving provides the following outcomes: where a 1 ; b 0 ; b 1 ; λ and μ are free parameters.
where a 0 ; b 0 ; λ and μ are free parameters.
Insert the values appeared in (3.1.6) and (3.1.7) in the solution (3.1.5) provide the following expressions for exact analytic solutions: The expressions (3.1.8) and (3.1.9) along with (2.2.7)-(2.2.9) make available the following closed form traveling wave solutions in terms of hyperbolic function, trigonometric function and rational function:

The nonlinear space-time fractional STO equation
Consider the nonlinear space-time fractional STO equation Using the complex fractional transformation Eqn (3.2.1) reduces to the following fractional order ordinary differential equation with respect to the variable ξ: ) creates a polynomial in ðD α ξ G=GÞ whose coefficients assigning to zero and solving yields the outcomes: where b 0 ; b 1 ; k; c; β and λ are all arbitrary constants.
where b 0 ; k; c; β and λ are all unknown parameters.
Utilizing the values available in (3.2.5) and (3.2.6) in (3.1.5) provide the following expressions for analytic solutions: The expressions (3.2.7) and (3.2.8) along with (2.2.7)-(2.2.9) make available the following closed form traveling wave solutions in terms of hyperbolic function, trigonometric function and rational function: 3.2.1 Solution 1. When λ 2 − 4μ > 0;

The nonlinear space-time fractional KPP equation
The nonlinear space-time fractional KPP equation is The fractional complex transformation Applying the homogeneous balance method to Eqn 3) forms a polynomial in ðD α ξ G=GÞ whose coefficients assigning to zero and solving gives up the following outcomes: where b 1 ; k; w; λ and μ are all unknown parameters.
The physical appearance of solutions to FNLEEs bears great importance to depict different phenomena arisen in various fields of nature in real world. This paper consists of some fresh and general solutions among which few are graphically brought up.  Physical appearance of solution (3.3.7) for λ ¼ 4; μ ¼ 3;

Conclusion
The core aim of this study is to make available further general and fresh closed form analytic wave solutions to the nonlinear space-time fractional PKP equation, the nonlinear space-time fractional STO equation and the nonlinear space-time fractional KPP equation through the suggested rational fractional ðD α ξ G=GÞ-expansion method. The offered method has successfully presented attractive solutions to the considered equations and shown its high performance. So far we know the achieved solutions are not available in the literature and p ¼ 0:5 within the interval −10 ≤ x; t ≤ 10 Solutions to FNLEEs in mathematical physics might create a milestone in research area to analyze the physical structure and behavior of the real life events that correspond to the fractional related models. Therefore, it may be claimed that the rational fractional ðD α ξ G=GÞ-expansion method in deriving the closed form analytical solutions is simple, straightforward and productive. This method might be taken into account for further implementation to investigate any FNLEEs arising in various fields of applied mathematics and mathematical physics. The obtained solutions in terms of trigonometric function, hyperbolic function and rational function containing many free parameters are claimed to be fresh and further general which will take place in the literature.