Mohmmd Asif Arefin , Md.Au Seed , M.Ali Akr , M.Hfiz Uddin ,
a Department of Mathematics, Jashore University of Science and Technology, Jashore, 7408, Bangladesh
b Department of Applied Mathematics, University of Rajshahi, Rajshahi, 6205, Bangladesh
ABSTRACT The (2 + 1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis (CD) and fractional potential Kadomstev-Petviashvili (PKP) equation.It can be modeled according to the Hamiltonian structure, the lax pair with the non-isospectral problem, and the pain level property.The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves,demonstrate the propagation of waves in dissipative and nonlinear media, and reveal the propagation of waves in dissipative and nonlinear media.In this paper, we have established further exact solutions to the nonlinear fractional partial differential equation (NLFPDEs), namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable ( G ′ /G, 1 /G )-expansion method.As far as trigonometric, hyperbolic, and rational function solutions containing parameters are concerned, solutions are acquired when unique characteristics are assigned to the parameters.Subsequently, the solitary wave solutions are generated from the solutions of the traveling wave.It is important to observe that this method is a realistic, convenient, well-organized,and ground-breaking strategy for solving various types of NLFPDEs.
Keywords:Two variable ( G ′ /G, 1 /G )-expansion method Exact solution Traveling wave solutions Solitary wave solutions The space-time fractional CD equation The space-time fractional PKP equation
Fractional calculus and consequently the fractional-order nonlinear partial differential equations have attracted the attention of many specialists due to their significance in portraying the inward component of the idea of this present reality.Non-linear fractional differential equations (NLFDEs) have been the subject of much research recently.Advances have been made in the fractional calculus theorem itself and the use of such advances in several fields,namely fluid mechanics, applied mathematics, engineering, fractional dynamics, biology, fluid flow, control theory, signal processing, physics, solid-state physics, plasma physics, plasma wave, optical fiber, elective fiber, etc.The issue of travelling waves, which is highly mathematical, applies to every aspect of the study of wave motions in the physical world.Traveling waves and their breaking on beaches, flood waves in rivers, ocean waves from storms, ship waves on water, and free oscillations of enclosed water such as lakes and harbors are among the mathematical and physical problems addressed.The nonlinear wave equation defines wave propagation in dispersive media such as liquid flow including gas bubbles, fluid flow in elastic tubes, rivers, lakes,and the ocean, as well as gravity waves in a corresponding domain and nonlinear wave motion rescaling.The use of this type of evolution equation for ocean wave motion and fluid flow research could be substantial.Consequently, many mathematical and analytical methods have been suggested for acquiring non-linear fractional differential equation solutions, such as the following methods: Adomian decomposition [1-3] , extended tanh-function[4 , 5], the homotopy analysis [6 , 7], variational iteration algorithm-1  , variational iteration algorithm-2  , the homotopy perturbation [10 , 11], differential transformation [12 , 13], local meshless[14 , 15], q-homotopy analysis transform [16-18] , generalized ( G ′ /G )-expansion  , finite element  , weak-form integral equation , the Darboux transformation  , exp-function [23 , 24], Sine-Gordon expansion  , modified simple equation  , fractional natural decomposition  , Kudryashov and modified Kudryashov , modified double sub-equation  , ( G ′ /G )- expansion [30-32] ,extended ( G ′ /G )- expansion [33 , 34], bilinear [35 , 36], two variable( G ′ /G, 1 /G )-expansion [37-41] , modified Sine-Gordon equation  ,and several more.
Mohyud-Din et al. established an analytical solution for the space-time fractional CD equation and space-time fractional PKP equation through the fractional sub-equation method, Cai et al. employed an exact and analytical solution to the spacetime CD equation or breaking soliton equation, taking into account the modified F-expansion method.Guner and Ozkan  recently studied the exact traveling wave solutions to the space-time fractional CD equation using the ansatz, exp-function and the ( G ′ /G ) -expansion methods.In addition, by applying the homogenous balance method, Bekir et al. investigated a new exact and explicit traveling wave solution for the PKP and CD equations via the homogenous balance method.Armed with the Tanh-Coth and Tan-Cot method, Anwar et al. searched the soliton solution to the space-time fractional PKP equation and space-time fractional CD equation.They also employed the Ansatz method.The latest exact traveling wave solution for the space-time non-linear fractional PKP equation was studied by Zayed et al. using the( G ′ /G ) -expansion method.It is important to note that the spacetime-fractional PKP equation and space-time fractional CD equation are not investigated by the two variable ( G ′ /G, 1 /G ) -expansion method which has only been recently developed.The purpose of this article is to build some advanced and exact solutions to the earlier mentioned models.The method described is practical, effective, and simple to compute for investigating the exact solutions to NLFDEs.
The rest of this article is structured as follows: In section 2 ,the necessary definitions and fundamental tools are given, while the two variable ( G ′ /G, 1 /G )-expansion is described in section 3 .In section 4 , we evaluate the exact solution to the CD and PKP equation by the suggested method and in section 5 graphical representation and discussion are illustrated.A comparison of the results appears in section 6 and section 7 concludes the paper.
The modified Riemann-Liouville derivative was developed by Jumarie (2006).First, we include some concepts and axioms that have been used in our analysis of this kind of fractional derivative.Assuming a continuous function is f : R → R, x → f(x ) the modified Riemann-Liouville derivative of order αof the Jumarie is defined by:
For the modified Riemann-Liouville derivative, those significant properties can be shown as follows:
where a and bare constants.
The direct outcome of (2.4) and (2.5) is
In this section, the following basic principles that help to define the two variable ( G ′ /G, 1 /G )-expansion method are discussed.Let’s assume an ordinary differential equation (ODE) of second order:
where φ= G ′ /G, ψ = 1 /G , then we obtain
Equation (3.1) provides hyperbolic, trigonometric and rational function solutions for the values of λ, when λ＜ 0 , λ＞ 0 and λ=0 .
Case 1 :For λ＜ 0 , the general solution of the equation (3.1) is obtained as follows:
Therefore, we get
where σ= A1-A2.
Case 2 :For λ＞ 0 , we obtain the subsequent solution of the equation (3.1)
Accordingly, it yields
where σ= A 1 + A 2 .
Case 3 :For λ= 0 , we attain the general solution of the equation(3.1)
As a result, it provides
where the arbitrary constants are A 1 and A 2 .
Consider the NLFDE in general form as follows:
u ( x, y, t ) , appear in (3.9) is an undefined function of spatial variable x, y and temporal variable t, P is a polynomial of u ( x, y, t )and its partial derivatives, involving maximal order of nonlinear terms and largest order of derivatives.
The vital steps of the two variable ( G ′ /G, 1 /G ) -expansion method are written in more detail below:
Step 1:Suppose the travelling wave transformation
where k1and k2and c are non-zero arbitrary constants.ξiscalled the wave variable.This allows the temporary variable tand the spatial variables x and y to be combined into a single variable.Using ξ, the nonlinear fractional equation (3.9) is converted into an integer-order ordinary differential equation as written below:
where R represents a polynomial in u (ξ) and its total derivative with respect to ξ.
Step 2:We consider that the solution of equation (3.3) can be represented in φand ψas a polynomial.
where ai, bi( i = 0 , 1 , 2 , ........., N) are constants which can be evaluated afterwards.
Step 3 :The positive integer Nthat points to the equation(3.12) was calculated by balancing the maximal order derivative in nonlinear and linear terms by using the homogenous balance in the equation (3.11) .
Step 4 :If equation (3.12) are replaced by equation (3.11) along with equations (3.2) and ( 3.4 ), equation (3.11) can be reduced to polynomials in φand ψ, where the degree of ψis equal to one or less than one.Every polynomial coefficient with a similar power to zero produces a system of algebraic equations which are resolved by applying computer algebra like Maple or Mathematica, yields ai, bi, μ, A1, A2and λvalues where λ＜ 0 , which give hyperbolic functional solutions.
Step 5:In a similar way, the values of ai, bi, μ, A1, A2and λare investigated, where the trigonometric and rational function solutions are respectively generated by λ＞ 0 , and λ= 0 .
In this section, the two variable ( G ′ /G, 1 /G )-expansion method serves to build the exact traveling wave solution to some nonlinear fractional PDEs, such as the space-time fractional CD equation and the space-time fractional PKP equation.In mathematical physics,these are very significant nonlinear fractional PDEs [45-48] .
4.1.The space-time fractional Calogero-Degasperis (CD) equation
The space-time fractional CD equation  can be written in the form:
The space-time fractional CD is significant because it describes various physical incidents in nonlinear science and engineering.It was first devised by Calogero and Degasperis  and this equation is utilized to narrate the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y -axis with a long wave along the x -axis.The non-isospectral problem of the Lax pair, the Hamiltonian structure, and the property of the pain level were also discussed.
For the CD equation, we consider the traveling wave transformation
Here, cis the traveling wave velocity.Equation (4.1) can be reduced to the following integer order of ordinary differential equation (ODE) applying traveling wave transformation equation (4.2) :
Integrating equation (4.3) with zero constant, we obtain
Balancing the nonlinear and linear terms of the maximal order derivative in equation (4.4) , provides N = 1 , so the solution (4.3) is transformed to the presentation below:
where a 0 , a 1 and b 1 are unknowns to be determined.
Case 1 :For λ＜ 0 replacing solution (4.5) into (4.4) along with(3.2) and (3.4) and considering the coefficients to zero, we obtain algebraic equations set for a0, a1, b1, c, k1, k2as follows:
Unraveling the set of algebraic equations in (4.6) by Maple or Mathematica, provides:
Again, for set 2, the exact solution to space-time fractional CD equation is obtained, as following:
Since A1and A2are integral constants, it could be arbitrarily chosen.From those if we select A1= 0 , A20 and μ= 0 , we gain the following solitary wave solution
Again, we extract the solitary wave solution if we set A10 , A2= 0 and μ= 0 in the equation (4.10) .
Case 2 :Likewise, when λ＞ 0 , inserting solution (4.5) into (4.4)alongside with equations (3.2) , ( 3.6 ) and equating to zero, we reach a set of mathematical equations for k1, k2, a0, a1, b1and cfrom which we obtain the results:
The exact solution to space-time fractional CD equation is obtained as follows for set 1:
Again, for set 2, the exact solution to the space-time fractional CD equation is obtained, and takes the following form:
The solution of equation (4.3) completes the substitution of the results into equation (4.5)
4.2.The space-time fractional PKP equation
Here, we consider the space-time fractional potential Kadomtsev-Petviashvili equation [43 , 45] in the form:
For the PKP equation, we consider the travelling wave transformation
where cis the traveling wave velocity.The equation (4.20) can be reduced to the following integer order ordinary differential equation (ODE) applying the wave transformation (4.21).
Balancing linear and nonlinear terms of highest order in (4.22)provides N = 1 .Therefore, the solution of (4.22) is transformed into the following equation:
where a0, a1and b1are constants.
Solving the over-determined set of algebraic equations accumulated in (4.24) via Mathematica, provides
For set 1, the exact solution to the space-time fractional PKP equation is obtained and written in the following form:
Case 2 :Likewise, if λ＞ 0 , embedding equation (4.23) with equations (3.2) , ( 3.6 ) and ( 4.22 ) to zero coefficient, we obtain the subsequent mathematical equations for k1, k2, a0, a1, b1and c,whose solutions are as:
The exact solution to the space-time fractional PKP equation is obtained as follows for set 1
Case 3 :Finally, when λ= 0 , inserting (4.23) with (3.2), (3.8)and (4.22) and equating zero coefficient we achieve a system of mathematical equations whose solution is:
The solution of equation (4.22) completes the substitution of the results in equation (4.23)
Fig.3.Sketch of the plan soliton wave of u 2 6 ( x, t ) , if A 1 = 1 , A 2 = 0 , μ= 0 , λ=-1 , k 1 = 1 , k 2 = 0 , a 0 = 0 , b 1 = 1 , α= 1 / 2 , 0 ≤t ≤10 , and -10 ≤x ≤10 .
It is interesting to see that some of the obtained solutions represent strong similarities with the solutions that were previously developed.Table 1 and Table 2 below provide, respectively, a correlation of the Gulnur Yel et al. and the Mohyud-Din et al. solutions.These are augmented by the solutions we have obtained.
Table 1 Comparison between Gulnur Yel et al. solutions and the obtained solutions
Table 2 Comparison between Mohyud-Din et al. solutions and the obtained solutions
For the above tables, the hyperbolic and trigonometric function solutions are comparable and are indistinguishable in the case that we set definite values of the arbitrary constants.It is important to understand that the time-space fractional PKP and CD equation’s traveling wave solutions u13( x, y, t ) , u16( x, y, t ) ,u18( x, y, t ) , u 19( x, y, t ) , u 111( x, y, t ) , u 23( x, y, t ) , u 26( x, y, t ) , u 28( x, y, t ) ,u29( x, y, t ) , u211( x, y, t ) , and u212( x, y, t ) are entirely new.It is also pertinent to point out that they were not covered in previous work.
In this article, we attained some new and more exact traveling wave solutions to the suggested equations via the two variable( G ′ /G, 1 /G )-expansion method along with the wave transformation.The results are obtained in terms of hyperbolic function, trigonometric function, and rational function solutions containing parameters, when special values are assigned to the parameters, and then the solitary wave solutions are created from the traveling wave solutions.The graphical representation and comparison between our achieved solutions and existing literature are successfully drawn.The achieved solutions for these equations make it possible to examine the (2 + 1) dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis.It also has the ability to explore the Lax pair’s non-isospectral problem,the Hamiltonian structure, and the pain level property.As well,it might be used to screen a variety of NLFPDEs in mathematical physics and coastal engineering.The obtained solutions can be used to investigate the hydrodynamics of wide channels or open seas of finite depth, the propagation of shallow-water waves with various dispersion relationships, the travel of shallow-water waves,and the propagation of waves in dissipative and nonlinear media,as well as beachfront ocean and ocean engineering.We can infer from the results that the competence of the proposed method is very effective, efficient, convenient and can be applied to other NLFPDEs-related solutions.
Authorship contribution statement
Mohammad Asif Arefin :Data curation, Visualization, Resources, Writing-original draft.Abu Saeed :Writing, Conceptualization, Methodology, Software, Validation.M.Ali Akbar :Software, Methodology, Writing-review editing.M.Hafiz Uddin :Formal analysis, Investigation, Writing-review editing, Supervision,Project administration.
Declaration of Competing Interest
The authors declare they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors would like to thank the anonymous referees and editor of the journal for their insightful comments and suggestions for improving the article.