Mohammad Asif Arefin ,M.AyeshKhatun ,M.Hafiz Uddin ,MustafInc

a Department of Mathematics,Jashore University of Science and Technology,Jashore 7408,Bangladesh

b Department of Computer Engineering,Biruni University,Istanbul,Turkey

c Department of Mathematics,Science Faculty,Firat University,Elazig,Turkey

d Department of Medical Research,China Medical University Hospital,China Medical University,Taichung,Taiwan

Keywords:Riemann-Liouville fractional derivative Space-time fractional (2+1)-dimensional dispersive long wave equation Approximate long water wave equation Wave transformation The two-variable ( G ′ /G,1 /G)-expansion method

ABSTRACT This work aims to construct exact solutions for the space-time fractional (2+1)-dimensional dispersive longwave (DLW) equation and approximate long water wave equation (ALW) utilizing the twovariable ( G ′ /G,1 /G)-expansion method and the modified Riemann–Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we produce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial differential equations (NLFPDEs) is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

1.Introduction

Many applications of fractional calculus (FC) can be found in displaying,signal preparing,material science,science,electromagnetism,medication,science,bioengineering and organic frameworks [1–4].In this way,numerous authors have been fascinated by learning fractional calculus and finding precise and productive techniques for comprehending NLFPDEs.All sections in the study of wave motions in the physical world are fascinated by the topic of travelling waves,which is highly mathematical and verified.The mathematical,as well as physical problems,deal with travelling waves and their breaking on beaches,with flood waves in rivers,with ocean waves from storms,with ship waves on water,with free oscillations of enclosed water such as lakes and harbors.The nonlinear wave equation describes the propagation of waves in dispersive media such as liquid flow containing gas bubbles,fluid flow in elastic tubes,rivers,lakes,and the ocean,as well as gravity waves in a comparable domain and spatio-temporal rescaling of the nonlinear wave motion.This type of evolution equation might have a significant impact on ocean wave motion and fluid flow research.

Many strategies have been produced for illuminating NLFPDEs,for example,nearby,the Darboux transformation method[5],weak-form integral equation method [6],Sine-Gordon equation expansion method [7],Adomian decomposition method [8,9 ],homotopy perturbation method [10,11 ],Lie’s invariant analysis method [12],the partial Lagrangian method [13],fractional sub equation method [14],Lie group analysis method [15],symmetrical differential operator method [16],q-homotopy analysis transform method [17],variational iteration method [18,19 ],the general Kudryashov method [20],Laplace decomposition method [21],improved(G′/G)-expansion method [22],the mass-spring damper system [23],the two-variable(G′/G,1/G)-expansion method[24–27],the finite element method [28],the generalized Riccati equation mapping method [29,30 ],the nonlinear least-square curve fitting technique [31],the perturbation-iteration algorithm[32],the collocation method [33],the lumped-mass method [34],theB-spline collocation method [35],the-expansion method [36],and so on.A clear and succinct method mentioned the two variable(G′/G,1/G)-expansion method,which is presented by [37]and exhibited that it is an influential method for looking at analytical solutions of NLDEs.

Fig.1.3D (left section) plot and contour (right section) plot represent kink type wave solution of u2 2 ( z,t) when σ=c=C2=1 ,C1=μ=0 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

Fig.2.3D (left section) plot and contour (right section) plot represent singular kink type wave solution of u2 3 ( z,t) when σ=c=C1=1 ,C2=μ=0 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

The space-time fractional DLW equations are very important to explain physical and mathematical phenomena in nonlinear science and ocean engineering.This equation describes the hydrodynamics of wide channels or open seas of finite depth,and in which the horizontal velocity and the wave elevation above the undisturbed water surface are included in which these are represented the time and the propagation plane,respectively.When the two independent variables become equal then the mention equation reduces to the (1+1)-dimensional system which can model the water wave propagation in certain infinitely long channels of finite constant depth and narrow width.The other equation namely space-time fractional ALW equations appear in hydrodynamics to demonstrate wave propagation in dissipative and nonlinear media,and they have a broad length of applications in coastal and ocean engineering.

Fig.3.3D (left section) plot and contour (right section) plot represent anti kink wave solution of v1 6 ( z,t) when a0=p0=C2=μ=k2=0 ,a1=p1=b1=p2=q1=C1=c=k1=1 ,α= ,0≤t≤10 and-10≤z≤10.

Fig.4.3D (left section) plot and contour (right section) plot represent kink type wave solution of u1 2 ( z,t) when C1=μ=a0=p0=k2=0 ,b1=c=k1=C2=1 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

The space-time fractional DLW equations have been researched for their precise diagnostic arrangements through the exp-function method by Zhang et al.[38],Yomba [39].Chen and Wang [40]have investigated the analytical solutions to the mention equations through modified extended Fan’s sub-equation method.Finally,Zheng [41]and Eslami et al.[42]have examined the same equation via(G′/G)-expansion method.Also,the proposed space-time fractional-coupled ALW equations have been studied for their exact analytic solutions through coupled fractional reduced differential transform method by Saha Ray [43].Guner et al.[44]have studied analytical and approximate solutions to the suggested equation by(G′/G)-expansion method,and so on.To our appreciation,the recommended condition has not been concentrated through the twovariable (G′/G,1/G)–expansion method.So the point of this investigation is to build up some fresh and advance broad precise solutions for the previously mentioned condition utilizing the double(G′/G,1/G)–expansion method.

Fig.5.3D (left section) plot and contour (right section) plot represent kink type wave solution of v1 2 ( z,t) when C1=μ=a0=p0=k2=0 ,b1=c=k1=C2=1 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

Fig.6.3D (left section) plot and contour (right section) plot represent kink type wave solution of u1 3 ( z,t) when C2=μ=a0=p0=k2=0 ,b1=c=k1=C1=1 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

The paper is decorated as: We have presented the definition and essential tools in Section 2.The two-variable (G′/G,1/G)-expansion method has been shown in Section 3.We have built up the specific answer for the proposed equation by the previously mentioned method in Section 4.In Section 5,we have presented a brief discussion and graphical representation.In Section 6,a comparison is given between obtained results and the previous results.At the end of the paper,the conclusion is given.

2.Definition and essential tools

Modified Riemann–Liouville fractional derivative was established by Jumarie in 2006.We can convert fractional differential equations (FDEs) into integer-order differential equations by using a fractional derivative,variable transformation,and certain accommodating methods.Letf:R→R,x→f(x)be a continuous function.The derivative of powerαis given as

Fig.7.3D (left section) plot and contour (right section) plot represent kink type wave solution of v1 3 ( z,t) when C2=μ=a0=p0=k2=0 ,b1=c=k1=C1=1 ,η=-1 ,α= ,0≤t≤10 and-10≤z≤10.

Fig.8.3D (left section) plot and contour (right section) plot represent periodic wave solution of u1 5 ( z,t) when C1=μ=a0=p0=k2=0 ,b1=c=k1=C2=η=1 ,α= ,0≤t≤10 and-10≤z≤10.

The modified Riemann–Liouville derivative has two or three axioms,and the remaining four celebrated requirements are presented as:

whereverαandbstands for constants,and

Which are the immediate results of

This holds for non-differentiable functions.The Eqs.(2.3) and(2.4),u(x)is differentiable but non-differentiable in Eq.(2.5) among the over Eqs.(2.3)–(2.5).The function u(x)is non-differentiable and f(u)is differentiable in Eq.(2.4) and non-differentiable in Eq.(2.5).So the explanation Eqs.(2.3)–(2.5)should be applied closely.

Fig.9.3D (left section) plot and contour (right section) plot represent periodic wave solution of u1 6 ( z,t) when a1=1 ,a0=0 ,p0=0 ,q2=1 ,p2=1 ,q1=1 ,C1=1 ,c=σ=k1=1 ,C2=μ=0 ,k2=0 ,α= ,0≤t≤10 and-10≤z≤10.

Fig.10.3D (left section) plot and contour (right section) plot represent kink wave solution of u1 7 ( z,t) when a0=p0=C2=μ=k2=0 ,a1=p1=b1=p2=q1=C1=c=k1=1 ,α= ,0≤t≤10 and-10≤z≤10.

3.The two-variable ( G ′ /G,1 /G)-expansion method

The two-variable(G′/G,1/G)-expansion method was proposed in [37]to get the specific traveling wave solution of NLFPDEs.Assume the standard second-order differential equation

also,the accompanying relations

Subsequently,this gives

The solution for Eq.(3.1) build uponηas likeη〈 0,η〉 0,andη=0.

Forη＜0,the complete solution of Eq.(3.1) is

Taking into account that we get

whereσ=C12-C22.

On the off chance thatη＞0,the solution for Eq.(3.1) will be as following form;

Considering that we acquire

While,η=0,the overall solution for condition (3.1) will be as like as

Taking into account that we acquire

hereC1andC2stand for constants and those are arbitrary.

Suppose that the general NLFDE is as form

At this point,u denotes an undiscovered spatial secondary functionyand transient subsidiarytand speaks to a polynomial of u(y,t)and nonlinear terms of maximum order,and the most maximum order of derivatives are connected in its derivatives.

Step 1: Firstly,take travelling wave transformation

where l andmare non-zero abstract constant.

By applying (3.11) into (3.10),it is reworked as:

Ordinary derivative of u with respect toξis represents prime here.

Step 2:Take the solution of Eq.(3.3) have been uncovered as polynomial inφandψo(hù)f the endorse type:

here,ai,biare stand for constants to be calculated lately.

Step 3: The maximal number of derivatives in linear and nonlinear terms appearing by homogeneous equilibrium in Eq.(3.12) fixed the positive integer numberNwhich decides the Eq.(3.13).

Step 4: Subbing (3.13) into (3.12) alongside (3.3) and (3.5) it diminishes to a polynomial inφandψ,having the degree one.Differentiating the polynomial of comparable terms with zero gives a blueprint of logarithmic equations that are analyzed by utilizing computational programming produces the assessments ofai,bi,μ,C1,C2andηwhereη＜0 which give exaggerated function courses of action.

Step 5: Likewise,we investigate the assessments ofai,bi,μ,C1,C2andηwhereη＞0,andη=0 which are giving trigonometric and rational function results correspondingly.

4.Formulation of exact solution

4.1.Perfect solutions of the fractional DLW equations in space and time

We have

which describes the hydrodynamics of wide channels or open seas of finite depth,and in whichu′(x,z,t)is the horizontal velocity,v′(x,z,t)represents the wave elevation above the undisturbed water surface,tand(x,z),respectively,stand for the time and the propagation plane.Whenz=x,system (4.1) reduces to the(1+1)-dimensional system which can model the water wave propagation in certain infinitely long channels of finite constant depth and narrow width.Eq.(4.1) was first procured by [42]utilizing a similarity equation for a ’’powerless’’ Lax pair.

Introduce the following fractional transformation

Substituting Eq.(4.2) in Eq.(4.1),we have

Applying the homogeneous equilibrium principle to Eq.(4.3a) and (4.3b) we getN1=1 andN2=2.By takingN1to be 1 andN2to be 2 in Eq.(4.3a) and (4.3b),we develop the form of the proposed solution of Eq.(4.1) in the following manner.

wherever,p0,p1,p2,q1andq2are constant to be resolved.

Case-1: Substituting Eq.(4.4a) and (4.4b) into (4.3a) and (4.3b)using (3.3) and (3.5) forη＜0,plucking all the terms with the equivalent power of(G′/G,1/G)together and a set of algebraic equations is obtained by reducing each coefficient to zero.Solving these equations yielded the following results.

Substituting these values into (4.4a) and (4.4b),we get the solution for the fractional (2+1)-dimensional (DLW) Eqs.(4.1) as the structure

SinceC1andC2are arbitrary constants.If we selectC2≠0 andC1=μ=0 in Eq.(4.5a) and (4.5b),solitary wave solution is obtained in the following manner

By choosingC1≠0 andC2=μ=0 in Eq.(4.5a) and (4.5b),we get the solitary wave solution

Case 2: Forη＞0,substituting Eq.(4.4a) and (4.4b) into (4.3a)and (4.3b) using (3.3) and (3.7) We achieve the resultant conclusion by solving mathematical problems with computer-based math like Maple.

Substituting these values into (4.4a) and (4.4b),we get the solution for the fractional (2+1)-dimensional (DLW) Eqs.(4.1) as the structure

It can be self-assertively chosen sinceC1andC2are arbitrary constants.In the event that we pickC1=μ=0 andC2≠0 in Eq.(4.6a) and (4.6b),then the following solitary wave solution are obtained

Acquire the solitary wave solution,by selectingC1≠0 andC2=μ=0 in Eq.(4.6a) and (4.6b),

Case 3:Likewise,for the same arrangement,whenη=0,setting Eqs.(4.4a) and (4.4b) into (4.3a) and (4.3b) using (3.3) and(3.9),we achieve the resultant conclusion by solving mathematical problems with computer-based math like Maple.

c=c,k1=k1,k2=0,a0=a0,a1=a1,b1=b1,q1=q1,q2=0,p0=p0,p2=p2,p1=p1.

We have get the rational function solution for the fractional(2+1)-dimensional (DLW) Eq.(4.1) by setting these values into(4.4a) and (4.4b),as the structure

4.2.Perfect solutions of the fractional-coupled ALW equations in space and time

We have

These equations appear in hydrodynamics to demonstrate wave propagation in dissipative and nonlinear media,and they have a broad length of applications in coastal and ocean engineering.They are also recommended for problems involving water leakage in the porous subsurface stratum.

Introduce the following fractional transformation

Applying Eq.(4.8) in Eq.(4.7),we have

From Eq.(4.10a),we obtain

Surrogating Eq.(4.11) in Eq.(4.10b)

Fig.11.3D (left section) plot and contour (right section) plot represent kink type wave solution of u2 5 ( z,t) when C2=μ=0 ,η=σ=C1=c=1 ,α= ,0≤t≤10and-10≤z≤10.

Fig.12.3D (left section) plot and contour (right section) plot represent kink type wave solution of u2 6 ( z,t) when C1=μ=0 ,η=σ=C2=c=1 ,α= ,0≤t≤10 and-10≤z≤10.

We acquire 2+m=3m ?m=1 by using the homogeneous equilibrium principle to Eq.(4.12).By considering m to be 2 in Eq.(3.13),we acquire the arrangement of the suggested solution of Eq.(4,16) in the following fashion.

wherevera0,a1andb1is constant.

Case-1: Substituting Eq.(4.13) into (4.12) using (3.3) and (3.5)forη＜0,plucking all the terms with the equivalent power of(G′/G,1/G)together and each coefficient is equalized to zero,resulting in a collection of algebraic equations..We have get the following results by solving these equations

We get the solution for the fractional ALW Eq.(4.7) by substituting the values of set 01 into (4.13),as the structure

Fig.13.3D (left section) plot and contour (right section) plot represent singular kink type wave solution of u2 7 ( z,t) when C1=1 ,C2=0 ,μ=0 ,c=1 ,0≤t≤10 and-10≤z≤10.

Fig.14.3D (left section) plot and contour (right section) plot represent kink wave solution of v1 7 ( z,t) when a0=p0=C2=μ=k2=0 ,a1=p1=b1=p2=q1=C1=c=k1=1 ,α= ,0≤t≤10 and-10≤z≤10.

SinceC1andC2are integration constants; their values can be freely chosen.In the event that we preferenceC1=μ=0 andC2≠0 in Eq.(4.14),solitary wave solution is obtained in the following fashion

By choosingC1≠0 andC2=μ=0 in Eq.(4.14),we get the solitary wave solution in the following manner

Case 2: Forη＞0,substituting Eq.(4.13) into (4.3a) using (3.3)and (3.7),we achieve the resultant conclusion by solving mathematical problems with computer-based math like Maple.

Substituting these values into (4.13),For set 01 we get the solution for the fractional ALW Eq.(4.7) as the structure

It can be self-assertively chosen sinceC1andC2are random constants.In the event that we pickC2≠0 andC1=μ=0 in Eq.(4.15),we get solitary wave solution

We obtain the solitary wave solution,if we preferenceC1≠0 andC2=μ=0 in Eq.(4.15).

Case 3: Likewise,for the same arrangement,whenη=0,setting Eqs.(4.13) into (4.12) using (3.3) and (3.9),by applying computer-based arithmetic,produces a solution to mathematical equations

Substituting the values of set 01into (4.13),we get the solution for the fractional ALW Eq.(4.7) as the structure

This one is essential to see that for the aftereffect of the constants given in set 2 for both in (cases 1–3),we obtain new and simpler solitary wave solutions.All solutions of the mentioned two equations namely DLW and ALW are fresh and furthermore general and have not been established in the previous solutions.Our achieved solutions are describing the hydrodynamics of wide channels or open seas of finite depth,the propagation of shallow-water waves with different dispersion relations,the travel of the shallow water wave,demonstrate the propagation of waves in dissipative and nonlinear media,and these are also extensively used in beachfront ocean and coastal engineering.

5.Brief discussion and graphical representation

Here,we give the graphical portrayal and actual clarification of some definite travelling wave solutions from this current investigation.The removed solutions of the proposed DLW and ALW equation have various types like trigonometric,hyperbolic,and rational function solutions.In this investigation,we plotted and examined three kinds of these arrangements.Fig.1 shows the kink type wave solution ofu22(z,t)for the values ofσ=c=C2=1,C1=μ=0,η=-1,α=,0≤t≤10 and-10≤z≤10.At the same process,Figs.4–7,10–12,and 14 shows the kink type wave solution ofu12(z,t)v12(z,t),u13(z,t),v13(z,t),u17(z,t),u25(z,t),u26(z,t)andv17(z,t)for different values of the constants.Kink wave is also known as topological or shock waves and it is a nonlinear wave where we have a transition from one stable state to another.Fig.3 shows the anti-kink type wave solution ofv16(z,t)for the values ofa0=p0=C2=μ=k2=0,a1=p1=b1=p2=q1=C1=c=k1=1,α=,0≤t≤10 and-10≤z≤10..In anti-kink wave velocity does not depend on the wave amplitude and which is opposite to kink type solution.Figs.2 and 13 shows the singular kink type wave solution ofu23(z,t)andu27(z,t)for the values ofσ=c=C1=1,C2=μ=0,η=-1,α=,0≤t≤10 and-10≤z≤10 andC1=1,C2=0,μ=0,c=1,0≤t≤10 and-10≤z≤10.Singular kink solution is another kind of travelling wave solution which comes from infinity as in trigonometry.Figs.8 and 9 shows periodic wave solution ofu15(z,t)andu16(z,t)for the values ofC1=μ=a0=p0=k2=0,b1=c=k1=C2=η=1,α=,0≤t≤10 and-10≤z≤10 anda1=1,a0=0,p0=0,q2=1,p2=1,q1=1,C1=1,c=σ=k1=1,C2=μ=0,k2=0,α=,0≤t≤10 and-10≤z≤10.Periodic wave is a wave whose amplitude periodically oscillates with the evolution of time.All figures in our article are offered in two configurations as 3D plot and contour plot.The acquired figures have been developed through calculation bundle program like Mathematica.

6.Results comparison

The derived solutions are compared in this section with Guner et al.[44]research outcomes.

Table 1

Table 1 Comparison between obtained results and Guner et al.[44]research outcomes.

The hyperbolic and trigonometric function solutions listed in the preceding table are comparable,and when the arbitrary constants are given definite values,they are indistinguishable.In a nutshell,it is important to understand that the travelling wave solutionsu11(z,t),u12(z,t),v12(z,t),u13(z,t),v13(z,t),u14(z,t),u15(z,t),v15(z,t),u16(z,t),v16(z,t),u17(z,t),v17(z,t),u21(z,t),u24(z,t),v24(z,t),u27(z,t)andv27(z,t)of the fractional ALW

equation and fractional (2+1)-dimensional DLW equations are all new and unique which are important to research of soliton theory that did not originate in the previous works.The other solution arising in set 02 produces also some new,unique,and further general solutions to the claim equation.The achieved solution obtained from the mention two equations is capable to clarify the above-stated phenomena.

7.Conclusion

We have studied space-time fractional (2+1)-dimensional dispersive long wave equations and space-time fractional-coupled ALW equation along with trigonometric,hyperbolic,and rational function solution containing parameters through Riemann-Liouville fractional derivative on the two-variable (G′/G,1/G)-expansion method in this work.The attained solutions may be functional to examine hydrodynamics of wide channels or open seas of finite depth,the propagation of shallow-water waves with different dispersion relations,the travel of the shallow water wave,the propagation of waves in dissipative and nonlinear media.These are also extensively used in beachfront ocean and coastal engineering.We realized the solution of the nature of solution multiple soliton wave solution,kink wave,double soliton wave solution,single soliton solution,periodic wave solution,and singular kink wave solutions by using the proposed method.For the precision of the outcomes,the 3D and contour profiles were obtained by allocating different parametric preferences.The obtained results in this analysis were verified by putting them back into NLFPDEs and accurately found with computational software Maple.The suggested method is a reliable,powerful,momentous,and especially persuading technique.It will also be applied to other NLFPDE structures that appear in nonlinear sciences as a rule.In addition,it might be used to screen a variety of NLFPDEs in mathematical physics and coastal engineering.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

CRediT authorship contribution statement

Mohammad Asif Arefin: Software,Data curation,Writing–original draft,Formal analysis.M.Ayesha Khatun: Software,Data curation,Writing–original draft.M.Hafiz Uddin: Conceptualization,Writing–review & editing,Validation.Mustafa Inc:Investigation,Supervision,Writing–review & editing.

Acknowledgment

The authors would like to thank the anonymous referees for their insightful comments and suggestion for improving the article.