Parthkumar P.Sartanpara,Ramakanta Meher

Department of Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, 395007 India

Keywords: Zakharov-Kuznetsov equations q-Homotopy analysis Shehu transform method Shehu transform Caputo fractional derivatives

ABSTRACT The application of the q-homotopy analysis Shehu transform method (q-HAShTM) to discover the estimated solution of fractional Zakharov-Kuznetsov equations is investigated in this study.In the presence of a uniform magnetic field,the Zakharov-Kuznetsov equations regulate the behaviour of nonlinear acoustic waves in a plasma containing cold ions and hot isothermal electrons.The q-HAShTM is a stable analytical method that combines homotopy analysis and the Shehu transform.This q-homotopy investigation Shehu transform is a constructive method that leads to the Zakharov-Kuznetsov equations,which regulate the propagation of nonlinear ion-acoustic waves in a plasma.It is a more semi-analytical method for adjusting and controlling the convergence region of the series solution and overcoming some of the homotopy analysis method’s limitations.

1.Introduction

Many researchers investigated the Nonlinear propagating dustacoustic solitary waves (DASWs) in a heated magnetized dusty plasma containing various sizes and mass negatively charged dust particles,isothermal electrons,high-and low-temperature ions.The Zakharov-Kuznetsov equation for the first-order perturbed potential was obtained using an adequate normalization of the hydrodynamic and Poisson equations.As the wave amplitude rose,the width and velocity of the solitons deviated from the predictions of the Zakharov-Kuznetsov equations.A linear inhomogeneous type Zakharov-Kuznetsov equation for the second-order perturbed potential was devised to explain the soliton of higher amplitude,whereas the re normalization method was used to provide stationary solutions to both equations.For the study of vortices in geophysical flows,The Zakharov-Kuznetsov equation was a beneficial model equation.The ZK equation,the electrostatic-acoustic pulses are analyzed in magnetized ions,is one of the notorious variations of the Korteweg-de Vries equations.The ZK equations are the investigation of coastal waves on the bases of ocean [1–4].ZK equation was initially introduced while studying weak nonlinear acoustic ion waves,which attract losses of two-dimensional ions.In the current work,we look at the following fractional-order ZK equation:

where,υ=υ(x,y,t),α∈(0,1] is parameter which defines the fractional order structure andCi(i=1, 2, 3) are arbitrary constants[5].ai(i=1, 2, 3) are integers and because ofai,the construction of weak nonlinear acoustic ion vibrations in a plasma comprising cooling,hot exothermic electrons in the systematic electric field is possible [6].

Leibniz first mentioned the name of fractional calculus;the well-known German mathematician sent a letter to other wellknown mathematicians from France named L’Hopital in the last decade of the 17th century.In the research article,fractional calculus was first introduced by Abel [7].Throughout time other great mathematicians Liouville [8,9],Reimann [10],Caputo [11] and many others [12–16] contributed their huge knowledge about the field.Fractional calculus has always been getting lots of attention by researchers due to its application in real-world physics problems and almost every engineering branch.There are many approaches available for the solution of nonlinear and fractional order differential equations [17–20].The solution of Fractional order ZK equation(FZK) was illustrated by many techniques as fractional iteration method (FIM) [21],perturbation-iteration algorithm and continues power series technique [6],New iterative Sumudu transform method(NISTM)[22],variational iteration method(VIM) [23],homotopy perturbation transform method(HPTM) [5],Laplace Adomian decomposition method(LADM) [24],fractional natural decomposition method(FNDM) andq-homotopy analysis transform method(q-HATM) [5],optimum homotopy asymptotic method(OHAM) [26],Homotopy analysis fractional Sumudu transform method(HAFSTM)[27],natural transform decomposition method(NTDM) [29] and Elzaki transform iterative method(ETIM) [28].All these techniques are providing an approximate solution for fractional nonlinear problem.The homotopy analysis method (HAM) was first introduced by Liao [30–32] in 1992 with the application of fundamental differentiation and geometry concept i.e.called homotopy.HAM was very beneficial for finding solution of many linear,nonlinear and fractional differential equation based problems [33–41].HAM was handy to find the solution without using perturbations or linearisation,and the method required a lot of computational work and a faster computer.To overcome these limitations,in 2012,El-Tawil and Huseen proposed a modified technique of HAM calledq-Homotopy Analysis Method(q-HAM) [42,43].q-HAM introduced a parameterq∈to control the convergence of series solution.As it was complex to find a solution of fractional order differential equation using,Singh et al.[44] introduced a new technique calledq-HATM,which was a combination ofq-HAM and Laplace transform in 2016.Next year,Singh et al.[45] used Sumudu transform instead of Laplace transform and provided a new approach to calledq-homotopy analysis Sumudu transform method(q-HASTM).Jena and Chakraverty [46] came up with the combination ofq-HAM and Aboodh transform and introduced a method calledq-homotopy analysis Aboodh transform method(q-HAATM) in 2019.In 2021,q-homotopy analysis Elzaki transform method(q-HAETM) was introduced by Singh et al.[47].As all those transforms have some limitations and not having direct use to many mathematical and nonlinear problems,it was not directly applicable to some problems.In 2019,Maitama and Zhao [48],introduced a new transform called Shehu transform to overcome limitations of available integral transforms.Later,Maitama and Zhao applied the Shehu transform successfully with the Homotopy Analysis Method to solve multidimensional fractional diffusion equation [49].In this paper,we introduce the new homotopy technique calledq-homotopy analysis Shehu transform method(q-HAShTM),a combination of theq-Homotopy analysis method and Shehu transform.This method controls the convergence of the series solution because of two parametersnand.The proposed technique is a time-saving technique and less complex than other techniques,and the efficiency is better.

2.Preliminaries

Definition 2.1A real functiong(t) ;t>0,is said to be in the spaceC?,?∈R if there exist a real numberp>?,such thatg(t)=tpg1(t),whereg1(t)∈C[0,∞ ),and it is said to be in the spaceiffg(k)∈,k∈N.Definition 2.2The Riemann-Liouville fractional derivative [15,16] of orderα >0,of a functiong(t) is defined as

wherenis integer andn-1<α≤n.Definition 2.3The Riemann-Lioullie fractional integral operator [15,16] of orderα≥0,of a functiong(t)∈C?,and?≤-1 is defined as

Definition 2.4The derivative ofg(t) with fractional order in the sense of Caputo [16] is defined as

Definition 2.5The Shehu transform [48] of the functiong(t)∈Awith

is defined by

where,sanduare positive numbers.Here,we can conduct the following results using Eq.(2.4)

Definition 2.6IfS[g(t)]=G(s,u),then its inverse Shehu transform [48] is given by

Theorem 2.1The sufficient condition for the existence of Shehu transform..Ifthefunctiong(t)isapiecewisecontinuesfunctionineveryfiniteinterval0≤t≤η andofexponentialorderζ fort>η.ThentheShehutransformG(s,u)exists[48].

Proof.We can construct the following statement algebraically asη≥0

Here,functiong(t) is a piecewise continues function in every finite intervalt∈[0,η],then the existence of first integral is obvious.

Similarly for the second interval,let’s consider the following case,as functiong(t) is of exponential orderζfort>η.

The proof is complete.Definition 2.7IfS[g(t)]=G(s,u) andg(n)(t) is a derivative ofnth order of the functiong(t) then the Shehu Transform ofg(n)is given by [48]

Definition 2.8Ifg1(t) andg2(t) be in the setAandS[g1(t)]=G1(s,u) andS[g2(t)]=G2(s,u),then [48]

whereg1?g2is convolution of two functionsg1(t) andg2(t),which is defined as [48]

Definition 2.9If the Shehu transformS[g(t)]=G(s,u) then the Shehu transform of Riemann-Liouville fractional derivative is given by [48]

where,α> 0 andn-1 <α≤n.Definition 2.10If the Shehu transformS[g(t)]=G(s,u) then the Shehu transform of Riemann-Liouville fractional integral is given by Mishra and Pandey [27]

Definition 2.11If the Shehu transformS[g(t)]=G(s,u) then the Shehu transform of Caputo fractional derivative is given by Mishra and Pandey [27]

where,α> 0 andn-1 <α≤n.

3.The procedure of q-HAShTM

To have a better understanding about the q-homotopy analysis Shehu transform method,let us consider the following nonlinear partial differential equation with fractional orderα∈(n-1,n]

Upon applying the Shehu transform on Eq.(3.1),it obtain

Simplifying Eq.(3.2),it finds

The Nonlinear operator can be defined as

As the value ofqincreases from 0 to,the solutionω(x,y,t;q)converges from the initial assumptionυ0(x,y,t) to the exact solutionυ(x,y,t).

On expandingω(x,y,t;q) in the Taylor series with respect toq,we obtain

whereυm(x,y,t) stands for

Atq=,if the choice of auxiliary linear operator,υ0(x,y,t),n,is proper than Eq.(3.7) converges to the one of the solution of Eq.(3.1)

Now,upon dividing Eq.(3.5) bym!,keepingm-times differentiation with respect toqand substitutingq=0,we obtain the deformation equation ofmth-order as follows

Upon applying the inverse Shehu transform on Eq.(3.11),

and solving the Eq.(3.15),we getυm(x,y,t) (form=1, 2, ···).

Thus the approximate series solution can be written as

Hence from Eq.(3.16),it is apparent that the series solution will surely be convergent in the presence ofthat controls the convergence of the series solution.

4.Convergence and absolute error analysis of the method

Theorem 4.1Convergence analysis of the method..Iftheapproximateseriessolution

isconvergestoχ(x,y,t),wheremth-orderdeformationEq.(3.11)basedonEqs.(3.12)and(3.14)generates υm(x,y,t),thenχ(x,y,t)mustbeasolutionoftheproblem Eq.(3.1).

Proof.Let us define

Then we have

From (3.11),we obtain

which yeilds,asH(x,y,t) andare non-zero and using linearity condition of Eq.(3.4),we obtain

Similarly from Eq.(3.14),we get

Thus from Eqs.(4.3) and (4.4),we can write

Hence,Eq.(4.5) proves thatχ(x,y,t) is the convergent solution of problem Eq.(3.1).□

Theorem 4.2Absolute error analysis of the method..Letυ(x,y,t)betheapproximatesolutionof.Assumethatthereexists arealnumberp∈(0,1)suchthat‖υk+1(x,y,t)‖≤p‖υk(x,y,t)‖,for?k,thenthemaximumabsoluteerroris

Proof.As the series solution is finite,so we can write

Thus,the proof is complete.□

5.Numerical applications

Example 5.1.Let us consider the Fractional Zakharov-Kuznetsov (2,2,2) equation [24] :

subject to the initial conditions

whereξis an arbitrary constant.The exact solution of the problem Eq.(5.1) forα=1 is

For solvng Eq.(5.1) using q-HAShTM,Let the initial approximation be

Upon transforming Eq.(5.1) using Shehu transform and using the condition (5.2),we can reach to the following expression

The nonlinear operator can be defined as

According to the method,the deformation equation of ordermis,

Upon transforming Eq.(5.7) using the inverse formula of Shehu Transform,we obtain

Upon using the initial assumption from Eq.(5.4),the first few terms of the solution can be expressed as following:

Thus the approximated series solution can be expressed as following:

Example 5.2.Let us consider the Fractional Zakharov-Kuznetsov (3,3,3) equation [24] :

subject to the initial conditions

whereξis an arbitrary constant.The exact solution of the problem Eq.(5.11) forα=1 is

For solving Eq.(5.11) using q-HAShTM,the initial approximation can be expressed as

Upon transforming Eq.(5.11) using Shehu transform and using condition (5.12),the following expression can be reached

The nonlinear operator can be defined as

According to the proposed method,the deformation equation of ordermis,

Upon transforming Eq.(5.17) using the inverse formula of Shehu transform,we obtain

Using initial assumption from Eq.(5.14),the first few terms of the solution are as following:

Thus the approximated series solution can be expressed as following:

6.Results and discussion

This section discusses the numerical solution of the fractional Zakharov-Kuznetsov equations.Figure 1 represents the graphical simulation for Ex.5.1.Figure 1 (a) and (b) are representing the graphical representation of the solution obtained byq-HAShTM and exact solution of Ex.5.1,respectively.Figure 1 (c) and (d) represents the plots ofq-HAShTM solution withα=0.5 andα=0.75,respectively.The analysis to different fractional points have been portrayed for Ex.5.1.Figure 2,is response to the solution obtained by distinct value ofα.Fig.3,represents the solution at different values of.Forn=1 andn=2,-curves as shown in Fig.4.Table 1 provides the numerical solution of Ex.5.1,at distinct value ofαand variablesx,y,andtatξ=0.001,=-1 andn=1.The comparative study of solution with different approaches as,VIM[23],FNDM [25],q-HATM [25] withq-HAShTM are presented in Table 2.Table 3 represents the terms approximations of the solution of Ex.5.1.

Table 1 q-HAShTM solution of Ex.5.1 for distinct value of α at ξ=0.001,=-1,n=1.

Table 1 q-HAShTM solution of Ex.5.1 for distinct value of α at ξ=0.001,=-1,n=1.

Table 2 Comparison between errors of q-HAShTM,VIM [23],FNDM [24] and q-HATM [24] for Ex.5.1 at ξ=0.001,=-1,n=1,α=1 and t=1.

Table 2 Comparison between errors of q-HAShTM,VIM [23],FNDM [24] and q-HATM [24] for Ex.5.1 at ξ=0.001,=-1,n=1,α=1 and t=1.

Table 3 Comparison between terms approximation solution of Ex.5.1 at α=1,=-1,n=1 and ξ=0.001.

Table 3 Comparison between terms approximation solution of Ex.5.1 at α=1,=-1,n=1 and ξ=0.001.

Fig.3. Plot of q -HAShTM solution of Ex.5.1 at ξ=0.001,α=1,n=1,x=0.5 and y=0.5 for different with respect to t.

Fig.4. -curves for Ex.5.1 when ξ=0.001.,x=0.5,y=0.5 and t=0.5 for two different values of n.

The solution of Ex.5.2 is presented graphically in Fig.5 (a),(b) of Fig.5 are the graphical simulations ofq-HAShTM solution atα=1 and exact solution,respectively and;(c) and (d) are the graphs plotted for solution of Ex.5.2 forα=0.5 andα=0.75,respectively.Figure 6 and 7 represent the graphical comparison at distinct values ofαand distinct values of,respectively.-curves forn=1 andn=2 are presented in Fig.8.Table 4 represents the numerical solutions of Ex.5.2,at different values ofαand variablesx,yandtatξ=0.001,=-1 andn=1.The comparison between the errors of different methods such as FNDM [25],q-HATM [25] withq-HAShTM is shown in Tables 5.Table 6 represents the terms approximations of the solution of Ex.5.2.

Table 4 q-HAShTM solution of Ex.5.2 for distinct value of α at ξ=0.001,=-1,n=1.

Table 4 q-HAShTM solution of Ex.5.2 for distinct value of α at ξ=0.001,=-1,n=1.

Table 5 Comparison between errors of q-HAShTM,FNDM [25] and q-HATM[25] for Ex.5.2 at ξ=0.001,n=1,α=1 and =-1.

Table 5 Comparison between errors of q-HAShTM,FNDM [25] and q-HATM[25] for Ex.5.2 at ξ=0.001,n=1,α=1 and =-1.

Table 6 Comparison between terms approximation solution of Ex.5.2 at α=1,=-1,n=1 and ξ=0.001.

Table 6 Comparison between terms approximation solution of Ex.5.2 at α=1,=-1,n=1 and ξ=0.001.

Fig.5. Surface of Ex.5.2 at ξ=0.001,=-1,n=1 and t=0.5.

Fig.6. Plot of q-HAShTM solution of Ex.5.2 at ξ=0.001,=-1,n=1,x=0.5 and y=0.5 for different α with respect to t.

Fig.7. Plot of q -HAShTM solution of Ex.5.2 at ξ=0.001,α=1,n=1,x=0.5 and y=0.5 for different with respect to t.

Fig.8. -curves for Ex.5.1 when ξ=0.001,x=0.5,y=0.5 and t=0.5 for two different values of n.

7.Conclusion

A robust semi-analytical method,namelyq-homotopy analysis Shehu transform approach,is applied to find the solution of Fractional Zakharov-Kuznetsov (2,2,2) and (3,3,3) equations.The graphical and numerical simulations are presented to understand the behaviour of the problem for both of the solutions.As two parameters ofq-HAShTM,namely,nandare very useful for managing the series solution’s convergence.The results show thatq-HAShTM is better than the classical methods such as VIM and are similar to other analytical methods such as FNDM andq-HATM.q-HAShTM works onq-HAM technique,which helps to obtain the controlled series solution.The Shehu transform helps solving the fractionalorder quickly,allowing the series solution to converge to the exact solution.Finally,it can be concluded that the approach ofq-Homotopy analysis Shehu transform method is very effective in understanding the nature of the nonlinear fractional partial differential equation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Ramakanta Meher