Abstract
In this article, a modified method called the Elzaki decomposition method has been applied to analyze time-fractional Zakharov–Kuznetsov equations. In this method, the Adomian decomposition technique and Elzaki transformation are combined. Two problems are investigated to show and validate the efficiency of the suggested method. It is also shown that the solutions achieved from the current producer are in good contact with the exact solutions to show with the help of graphs and table. It is observed that the suggested technique is to be reliable, efficient, and straightforward to implement for many related models of engineering and science.
1. 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. These 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 [1–4].
The Zakharov–Kuznetsov (ZK) equation is an extremely appealing model equation for investigating vortices in geophysical streams. The ZK problems show up in numerous regions of material science, implemented arithmetic, and designing. Specifically, it appears in the territory of quantum physics [5–9]. The 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 time-fractional 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 [18–23]. Consequently, special consideration has been given to discover solutions of fractional differential equations.
The 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. The 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 [30–36].
In this present work, the Elzaki decomposition technique is applied to investigate the result of the fractional-order ZK equation. The fractional derivatives are defined by the Caputo operator. The result of the given problems shows the validity of the suggested method. The 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. The given method is very helpful in solving other fractional-order of PDEs.
2. Basic Definitions
2.1. Definition
The fractional-order Riemann–Liouville , of a function , , is given as [27]
The operator of some properties:
For , ,
2.2. Lemma
The basic theory of the Elzaki transformation:
A new transform called the Elzaki transform defines the function exponential order that we found in the set , define by [27]
The finite number must be constant, and of infinite or finite, for a specified function in the set. The Elzaki transformation is defined throughout the following integral problem:
We can obtain the next solution from the explanation and the basic investigation
2.3. Theorem
The Elzaki Riemann–Liouville transform of the derivative can be defined as given if is the Elzaki transformation of [27]:
proof. Taking the Laplace transformation of we haveTherefore, the fractional-order Elzaki transform of is
2.4. Definition
Using Theorem 1, the fractional Caputo ET is provided as [18–20]
3. 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, and hencewhere is defined as
The nonlinearity of Adomian polynomials terms 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
4. Main Results
Example 1. Consider the two-dimensional Zakharov–Kuznetsov equation asand the initial condition iswhere is an arbitrary constant.
Taking Elzaki transform of equation (23), we haveApplying the inverse Elzaki transform, we haveUsing ADM procedure, we getwhere the nonlinear terms can be defined by Adomian polynomials in the above equations.Adomian polynomials are given asfor The few terms of the given methods areThe EDM result isFor , we haveIn Figure 1, the exact and the EDM solutions of problem 1 at 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 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 and and second one shows the different fractional-order graph with respect to and . Table 1 shows the different fractional-order absolute error.
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Example 2. Consider the three-dimensional Zakharov–Kuznetsov equation asand the initial condition iswhere is an arbitrary constant.
Taking Elzaki transform of equation (34), we haveUsing inverse Elzaki transformation,Applying the procedure of ADM, we getwhere the nonlinear terms can be defined by Adomian polynomials in the above equations.Adomian polynomials are given asfor The subsequent terms areThe EDM result isFor ,In Figure 3, the exact and the EDM solutions of Example 2 at are shown by Figures 3(a) and 3(b), respectively. From the given figures, it can be seen that both the EDM and exact solutions are in close contact with each other. Also, in Figures 3(c) and 3(d), the EDM results of problem 2 are investigated at different fractional-order 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 4, the first graph shows the two dimensions of exact and analytical solutions with respect to and the second one shows the different fractional-order graph with respect to .
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5. Conclusion
In this article, we investigated the time-fractional Zakharov–Kuznetsov equations using an Elzaki decomposition method. The given test examples illustrate the leverage and effectiveness of the suggested method. The obtained solutions are demonstrated by tables and graphs. The Elzaki decomposition method solution is in close contact with the actual result of the given problems. The 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
The 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
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
Pongsakorn Sunthrayuth and Farman Ali contributed equally to this work.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through Research Groups Program under grant number (R.G.P2./99/41).