The Numerical Investigation of Fractional-Order Zakharov–Kuznetsov Equations

Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala, University of Technology anyaburi (RMUTT), Pathumthani 12110, ailand Department of Software, Sejong Unversity, Seoul 05006, Republic of Korea Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan Department of Mathematics, Faculty of Science, AL-Azhar University, Assiut, Egypt Department of Mathematics and Statistics, College of Science, Taif University, P O. Box 11099, Taif 21944, Saudi Arabia Central Department of Mathematics, Tribhuvan University, Kritipur, Nepal


Introduction
Nonlinear fractional partial differential equations play important role in demonstrating different physical appearances identified with solid-state physics, fluid mechanics, chemical kinetics, population dynamics, plasma physics, nonlinear optics, protein chemistry, soliton theory, etc. ese nonlinear problems, just as their scientific arrangements, are of tremendous enthusiasm for suitable subjects. In many above-discussed science and engineering areas, the nonlinear problems perform a key factor in many phenomena. Differential equations demonstrate several frameworks and the majority of them are nonlinear [1][2][3][4]. e Zakharov-Kuznetsov (ZK) equation is an extremely appealing model equation for investigating vortices in geophysical streams. e ZK problems show up in numerous regions of material science, implemented arithmetic, and designing. Specifically, it appears in the territory of quantum physics [5][6][7][8][9]. e ZK problems administer the conduct of feebly nonlinear particle acoustic plasma waves, including cold particles and hot isothermal electrons within sight of a smooth magnetic field [10,11]. Solitary wave arrangements were produced by determining the nondirect higher order of broadened KdV conditions for the free surface removal [12]. By utilizing fractional strategy, the precise expository structures of some nonlinear advancement equations in numerical material science, to be specific, space timefractional Zakharov-Kuznetsov and modified Zakharov-Kuznetsov equations, were obtained [13]. It has been investigated in the past decades by many with the techniques such as new iterative Sumudu transform method [14], homotopy perturbation transform method [15], extended direct algebraic method [16], natural decomposition method, and q-homotopy analysis transform method [17].
In the last three decades, fractional differential conditions have picked up importance and ubiquity, mostly since its exhibited uses in various fields of material science and design. Numerous significant phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science, likelihood and measurements, electrochemistry of erosion, concoction physical science, and sign preparation are depicted in fractional differential equations [18][19][20][21][22][23]. Consequently, special consideration has been given to discover solutions of fractional differential equations. e investigation of these equations and their solutions has extraordinary enthusiasm for numerous specialists because of its different applications. To refer to a couple, Wazwaz [24] used the Adomian disintegration strategy as a dependable method for treating Schrodinger conditions. In Wazwaz [25], the variational emphasis technique was utilized to obtain specific solutions for both linear and nonlinear Schrodinger equations. Additionally, Shah et al. [26] utilized He's recurrence definition as a technique to look for Schrodinger equations arrangements. e arrangements decided to end up being in good concurrence with the outcomes decided in [24,25]. Notwithstanding, we mean to couple the Elzaki transform built up as late by Elzaki [27] with the commended technique for the 80th Adomian decay strategy [28,29]. Recently, many researchers obtained the results of FPDEs; interested readers can see [30][31][32][33][34][35][36].
In this present work, the Elzaki decomposition technique is applied to investigate the result of the fractional-order ZK equation.
e fractional derivatives are defined by the Caputo operator. e result of the given problems shows the validity of the suggested method. e solutions of the suggested technique are analyzed and shown with the help of the table and figures. Applying the current method, the results of time-fractional equations and integral-order equations are investigated. e given method is very helpful in solving other fractional-order of PDEs.

Definition.
e fractional-order Riemann-Liouville ρ > 0, of a function f ∈ C ı , ρ ≥ −1, is given as [27] (1) e operator of some properties: For h ∈ C ı , ρ ≥ − 1, ρ, β ≥ 0, and ρ > − 1, [18][19][20], e basic theory of the Elzaki transformation: A new transform called the Elzaki transform defines the function exponential order that we found in the set A, define by [27] A � h(I): 2 Complexity e finite number M must be constant, k 1 and k 2 of infinite or finite, for a specified function in the set. e Elzaki transformation is defined throughout the following integral problem: We can obtain the next solution from the explanation and the basic investigation

eorem.
e Elzaki Riemann-Liouville transform of the derivative can be defined as given if T(s) is the Elzaki transformation of (I) [27]: proof.
Taking the Laplace transformation of

The General Implementation of Elzaki Decomposition Technique
In this section, we present the Elzaki decomposition technique producer for fractional partial differential equation.
and the initial condition is Applying the Elzaki transform to equation (11), we get Using the Elzaki transform differentiation property, Now, Ψ(ξ, 0) � k(ξ) and hence where Ψ(ξ, I) is defined as e nonlinearity of Adomian polynomials terms N is defined as Putting equation (16) and (17) into (15), we have Now using EDM, we have Generally, we can write Taking the inverse Elzaki transform of equation (21), we have

Main Results
Example 1. Consider the two-dimensional Zakharov-Kuznetsov equation as and the initial condition is where η is an arbitrary constant. Taking Elzaki transform of equation (23), we have

Complexity
Applying the inverse Elzaki transform, we have Using ADM procedure, we get where the nonlinear terms can be defined by Adomian polynomials in the above equations.
Adomian polynomials are given as for ı � 0, 1, 2, . . . , Complexity 5 e few terms of the given methods are , e EDM result is cosh(ξ + ζ) 88400 cosh 6 (ξ + ζ) − 160200 cosh 4 (ξ + ζ) + 85170 cosh 2 (ξ + ζ) − 11903 I 3ρ Γ(3ρ + 1) For ρ � 1, we have In Figure 1, the exact and the EDM solutions of problem 1 at ρ � 1 are shown by Figures 1(a) and 1(b), respectively. From the given figures, it can be seen that both the EDM and exact results are in close contact with each other. Also, in Figures 1(c) and 1(d), the EDM solutions of problem 1 are investigated at different fractional-order ρ � 0.8 and 0.6. It is analyzed that time-fractional problem results are convergent to an integer order effect as time-fractional analysis to integer order. In Figure 2, the first graph shows the two dimensions of exact and analytical solutions with respect to ξ and I and second one shows the different fractional-order graph with respect to ξ and I. Table 1 shows the different fractional-order absolute error.

Conclusion
In this article, we investigated the time-fractional Zakharov-Kuznetsov equations using an Elzaki decomposition method. e given test examples illustrate the leverage and effectiveness of the suggested method. e obtained solutions are demonstrated by tables and graphs. e Elzaki decomposition method solution is in close contact with the actual result of the given problems. e figures show that the time-fractional solutions obtained have verified the convergence towards the integer order solutions. Moreover, the current technique is simple and straightforward as compared to other analytical techniques; the proposed method can solve other linear and nonlinear fractional-order partial differential equations.

Data Availability
e numerical data used to support the findings of this study are included within the article.

Disclosure
Pongsakorn Sunthrayuth and Farman Ali are the co-first authors.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.