# The Numerical Investigation of Fractional-Order Zakharov–Kuznetsov Equations

Authors: Pongsakorn Sunthrayuth; Farman Ali; A. A. Alderremy; Rasool Shah; Shaban Aly; Y. S. Hamed; Jeevan Katle
Journal: Complexity
 (2021)

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## 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.


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## Body

## 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–Liouvilleρ>0, of a function f∈Cı, ρ≥−1, is given as [27](1)Jρhξ=1Γρ∫0ξξ−1ρ−1hη∂η,ρ,ξ>0,Jρhξ=hξ.The operator of some properties:Forh∈Cı, ρ≥−1, ρ,β≥0,andρ>−1,(2)JρJβhξ=Jρ+βhξ,JρJβhξ=JρJβhξ,Jβξρ=Γρ+1β+ρ+1ξβ+ρ.


### 2.2. Lemma

Ifı−1<ρ≤ı,ı∈N and h∈Cı,ρ≥−1, then DρJρhξ=hξ [18–20],(3)DρJρhξ=hξ−∑ı=0m−1hı0ξıı!,ξ>0.The basic theory of the Elzaki transformation:A new transform called the Elzaki transform defines the function exponential order that we found in the setA, define by [27](4)A=hℑ:∑M,k1,k2>0,hℑ|<Meℑ/kı,ifℑ∈−1ı×0,∞.The finite numberM must be constant, k1 and k2 of infinite or finite, for a specified function in the set. The Elzaki transformation is defined throughout the following integral problem:(5)Ehℑ=Ts=s∫0∞hℑe−ℑ/sdℑ,ℑ≥0,k1≤s≤k2.We can obtain the next solution from the explanation and the basic investigation(6)Eℑn=m!sm+2,Eh′ℑ=Tss−sh0,Eh″ℑ=Tss2−h0−sh′0,Ehmℑ=Tssm−∑k=0n−1s2−m+khk0.


### 2.3. Theorem

The Elzaki Riemann–Liouville transform of the derivative can be defined as given ifTs is the Elzaki transformation of ℑ [27]:(7)EDρhℑ=s−ρTs−∑k=1mDρ−kh0,−1<m−1≤ρ<m.proof.
Taking the Laplace transformation ofh′ℑ=d/dℑhℑ, we have(8)LDρhℑ=sρTs−∑k=0m−1skDρ−k−1h0=sρTs−∑k=0m−1sk−1Dρ−kh0=sρTs−∑k=0m−1sk−2Dρ−kh0=sρTs−∑k=0m−11s−k+2Dρ−kh0=sρTs−∑k=0m−11sρ−k+2−ρDρ−kh0=sρTs−∑k=0m−1sρ1sρ−k+2Dρ−kh0LDρhℑ=sρTs−∑k=0m−11sρ−k+2Dρ−kh0.
Therefore, the fractional-order Elzaki transform ofhℑ is(9)EDρhℑ=s−ρTs−∑k=0msρ−k+2Dρ−kh0.


### 2.4. Definition

Using Theorem 1, the fractional Caputo ET is provided as [18–20](10)EDℑρgℑ=sρEgℑ−∑k=0ı−1s2−ρ+kgk0,whereı−1<ρ<ı.


## 2.1. Definition

The fractional-order Riemann–Liouvilleρ>0, of a function f∈Cı, ρ≥−1, is given as [27](1)Jρhξ=1Γρ∫0ξξ−1ρ−1hη∂η,ρ,ξ>0,Jρhξ=hξ.The operator of some properties:Forh∈Cı, ρ≥−1, ρ,β≥0,andρ>−1,(2)JρJβhξ=Jρ+βhξ,JρJβhξ=JρJβhξ,Jβξρ=Γρ+1β+ρ+1ξβ+ρ.


## 2.2. Lemma

Ifı−1<ρ≤ı,ı∈N and h∈Cı,ρ≥−1, then DρJρhξ=hξ [18–20],(3)DρJρhξ=hξ−∑ı=0m−1hı0ξıı!,ξ>0.The basic theory of the Elzaki transformation:A new transform called the Elzaki transform defines the function exponential order that we found in the setA, define by [27](4)A=hℑ:∑M,k1,k2>0,hℑ|<Meℑ/kı,ifℑ∈−1ı×0,∞.The finite numberM must be constant, k1 and k2 of infinite or finite, for a specified function in the set. The Elzaki transformation is defined throughout the following integral problem:(5)Ehℑ=Ts=s∫0∞hℑe−ℑ/sdℑ,ℑ≥0,k1≤s≤k2.We can obtain the next solution from the explanation and the basic investigation(6)Eℑn=m!sm+2,Eh′ℑ=Tss−sh0,Eh″ℑ=Tss2−h0−sh′0,Ehmℑ=Tssm−∑k=0n−1s2−m+khk0.


## 2.3. Theorem

The Elzaki Riemann–Liouville transform of the derivative can be defined as given ifTs is the Elzaki transformation of ℑ [27]:(7)EDρhℑ=s−ρTs−∑k=1mDρ−kh0,−1<m−1≤ρ<m.proof.
Taking the Laplace transformation ofh′ℑ=d/dℑhℑ, we have(8)LDρhℑ=sρTs−∑k=0m−1skDρ−k−1h0=sρTs−∑k=0m−1sk−1Dρ−kh0=sρTs−∑k=0m−1sk−2Dρ−kh0=sρTs−∑k=0m−11s−k+2Dρ−kh0=sρTs−∑k=0m−11sρ−k+2−ρDρ−kh0=sρTs−∑k=0m−1sρ1sρ−k+2Dρ−kh0LDρhℑ=sρTs−∑k=0m−11sρ−k+2Dρ−kh0.
Therefore, the fractional-order Elzaki transform ofhℑ is(9)EDρhℑ=s−ρTs−∑k=0msρ−k+2Dρ−kh0.


## 2.4. Definition

Using Theorem 1, the fractional Caputo ET is provided as [18–20](10)EDℑρgℑ=sρEgℑ−∑k=0ı−1s2−ρ+kgk0,whereı−1<ρ<ı.


## 3. The General Implementation of Elzaki Decomposition Technique

In this section, we present the Elzaki decomposition technique producer for fractional partial differential equation.(11)DρΨξ,ℑ+LΨξ,ℑ+NΨξ,ℑ=qξ,ℑ,ξ,ℑ≥0,ı−1<ρ<ı,and the initial condition is(12)Ψξ,0=kξ.Applying the Elzaki transform to equation (11), we get(13)EDρΨξ,ℑ+ELΨξ,ℑ+NΨξ,ℑ=Eqξ,ℑ.Using the Elzaki transform differentiation property,(14)1sρEΨξ,ℑ−s2−ρΨξ,0=Eqξ,ℑ−ELΨξ,ℑ+NΨξ,ℑEΨξ,ℑ=s2Ψξ,0+sρEqξ,ℑ−sρELΨξ,ℑ+NΨξ,ℑ.Now,Ψξ,0=kξ and hence(15)EΨξ,ℑ=s2kξ+sρEqξ,ℑ−sρELΨξ,ℑ+NΨξ,ℑ,where Ψξ,ℑ is defined as(16)Ψξ,ℑ=∑ı=0∞Ψıξ,ℑ.The nonlinearity of Adomian polynomials termsN is defined as(17)NΨξ,ℑ=∑ı=0∞Aı,(18)Aı=1ı!dıdλıN∑ı=0∞λıΨıλ=0,ı=0,1,2,….Putting equation (16) and (17) into (15), we have(19)E∑ı=0∞Ψıξ,ℑ=s2kξ+sρEqξ,ℑ−sρEL∑ı=0∞Ψıξ,ℑ+∑ı=0∞Aı.Now using EDM, we have(20)EΨ0ξ,ℑ=s2kξ+sρEqξ,ℑ.Generally, we can write(21)EΨı+1ξ,ℑ=−sρELΨıξ,ℑ+Aı,ı≥1.Taking the inverse Elzaki transform of equation (21), we have(22)Ψ0ξ,ℑ=kξ+E−1sρEqξ,ℑ,Ψı+1ξ,ℑ=−E−1sρELΨıξ,ℑ+Aı.


## 4. Main Results

Example 1.
Consider the two-dimensional Zakharov–Kuznetsov equation as(23)DℑρΨ+Ψ2ξ+18Ψ2ξξξ+18Ψ2ξζζ=0,0<ρ≤1,and the initial condition is(24)Ψξ,ζ,0=43ηsinh2ξ+ζ,where η is an arbitrary constant.
Taking Elzaki transform of equation (23), we have(25)E∂ρΨ∂ℑρ=E−Ψ2ξ−18Ψ2ξξξ−18Ψ2ξζζ,1sρEΨξ,ζ,ℑ−s2−ρΨξ,ζ,0=E−Ψ2ξ−18Ψ2ξξξ−18Ψ2ξζζ.
Applying the inverse Elzaki transform, we have(26)Ψξ,ζ,ℑ=E−1s2Ψξ,ζ,0+E−1sρE−Ψ2ξ−18Ψ2ξξξ−18Ψ2ξζζ,Ψξ,ζ,ℑ=43ηsinh2ξ+ζ+E−1sρE−Ψ2ξ−18Ψ2ξξξ−18Ψ2ξζζ.
Using ADM procedure, we get(27)∑ı=0∞Ψıξ,ζ,ℑ=43ηsinh2ξ+ζ+E−1sρE−NΨξ−18NΨξξξ−18NΨζζξ,where the nonlinear terms can be defined by Adomian polynomials in the above equations.(28)NΨ=Ψ2=∑ı=0∞AıΨ,
Adomian polynomials are given as(29)A0=Ψ02,A1=2Ψ0Ψ1,A2=2Ψ0Ψ2+Ψ12,Ψ0ξ,ζ,ℑ=43ηsinh2ξ+ζ,Ψı+1ξ,ζ,ℑ=E−1sρE−∑ı=0∞AıΨξ−18∑ı=0∞AıΨξξξ−18∑ı=0∞AıΨζζξ,for ı=0,1,2,…,(30)Ψ1ξ,ζ,ℑ=E−1sρE−Ψ02ξ−18Ψ02ξξξ−18Ψ02ξζζ,Ψ1ξ,ζ,ℑ=−2249η2sinh3ξ+ζcoshξ+ζ−323η2sinhξ+ζcosh3ξ+ζE−1sρ+2Ψ1ξ,ζ,ℑ=−2249η2sinh3ξ+ζcoshξ+ζ−323η2sinhξ+ζcosh3ξ+ζℑρΓρ+1.
The few terms of the given methods are(31)Ψ2ξ,ζ,ℑ=E−1sρE−2Ψ0Ψ1ξ−182Ψ0Ψ1ξξξ−182Ψ0Ψ1ξζζ,Ψ2ξ,ζ,ℑ=12827η31200cosh6ξ+ζ−2080cosh4ξ+ζ+968cosh2ξ+ζ−79ℑ2ρΓ2ρ+1,Ψ3ξ,ζ,ℑ=E−1sρE−2Ψ0Ψ2+Ψ12ξ−182Ψ0Ψ2+Ψ12ξξξ−182Ψ0Ψ2+Ψ12ξζζ,Ψ3ξ,ζ,ℑ=−204881η4sinhξ+ζcoshξ+ζ88400cosh6ξ+ζ−160200cosh4ξ+ζ+85170cosh2ξ+ζ−11903ℑ3ρΓ3ρ+1.
The EDM result is(32)Ψξ,ζ,ℑ=Ψ0ξ,ζ,ℑ+Ψ1ξ,ζ,ℑ+Ψ2ξ,ζ,ℑ+Ψ3ξ,ζ,ℑ+⋯,Ψξ,ζ,ℑ=43ηsinhξ+ζ−2249η2sinh3ξ+ζcoshξ+ζ+323η2sinhξ+ζcosh3ξ+ζℑρΓρ+1+12827η31200cosh6ξ+ζ−2080cosh4ξ+ζ+968cosh2ξ+ζ−79ℑ2ρΓ2ρ+1−204881η4sinhξ+ζcoshξ+ζ88400cosh6ξ+ζ−160200cosh4ξ+ζ+85170cosh2ξ+ζ−11903ℑ3ρΓ3ρ+1+⋯.
Forρ=1, we have(33)Ψξ,ℑ=43ηsinh2ξ+ζ−ηℑ.
In Figure1, 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.Figure 1
(a) The exact solution figure ofΨξ,ℑ of Example 1. (b) The EDM solution figure of Ψξ,ℑ of Example 1. (c) The graph EDM result of Ψξ,ℑ at ρ=0.8 problem 1. (d) The graph EDM result of Ψξ,ℑ at ρ=0.6 problem 1.
(a)(b)(c)(d)Figure 2
(a) The exact and EDM solution figure ofΨξ,ℑ of Example 1. (b) The EDM solution figure of different fractional-order of ρ at Ψξ,ℑ of Example 1. (c) The exact and EDM solution figure of Ψξ,ℑ with respect to ℑ Example 1. (d) The EDM solution figure of different fractional-order of ρ with respect to ℑ of Example 1.
(a)(b)(c)(d)Table 1
EDM result for a different value ofρ when η=0.001 and absolute error of Example 1.
EDMξζℑρ=0.6ρ=0.8EDMρ=1Exact (E)AE (E)ρ=10.20.20.16.347977E −56.378034E −54.461135E −54.488458-52.7323-70.36.374898E −56.397867E −54.441928E −54.499518-55.76800-70.56.329478E −56.358477E −54.393236E −54.488588-58.92644-70.50.50.13.961810E −33.992936E −34.124545E −34.148347-32.3802-50.33.918638E −33.937942E −32.767287E −32.868188-31.00902-40.52.851320E −33.986888E −33.014248E −33.144685-31.31437-41.01.00.12.651681E −22.755798E −22.78556E −022.864629-27.9069-40.32.147458E −22.554775E −22.69884E −022.863881-21.65041-30.53.686988E −33.286775E −22.55842E −022.861074-23.02654-3Example 2.
Consider the three-dimensional Zakharov–Kuznetsov equation as(34)DℑρΨ+Ψ3ξ+2Ψ3ξξξ+2Ψ3ξζζ=0,0<ρ≤1,and the initial condition is(35)Ψξ,ζ,0=32ηsinh16ξ+ζ,where η is an arbitrary constant.
Taking Elzaki transform of equation (34), we have(36)E∂ρΨ∂ℑρ=E−Ψ3ξ−2Ψ3ξξξ−2Ψ3ξζζ,sρEΨξ,ζ,ℑ−s2−ρΨξ,ζ,0=E−Ψ3ξ−2Ψ3ξξξ−2Ψ3ξζζ.
Using inverse Elzaki transformation,(37)Ψξ,ζ,ℑ=E−1s2Ψξ,ζ,0+sρE−Ψ3ξ−2Ψ3ξξξ−2Ψ3ξζζ,Ψξ,ζ,ℑ=32ηsinh16ξ+ζ+E−1sρE−Ψ3ξ−2Ψ3ξξξ−2Ψ3ξζζ.
Applying the procedure of ADM, we get(38)∑ı=0∞Ψıξ,ζ,ℑ=32ηsinh16ξ+ζt+E−1sρE−NΨξ−2NΨξξξ−2NΨξζζ,where the nonlinear terms can be defined by Adomian polynomials in the above equations.(39)NΨ=Ψ3=∑ı=0∞ℬıΨ.
Adomian polynomials are given as(40)ℬ0=Ψ03,ℬ1=3Ψ02Ψ1,ℬ2=3Ψ02Ψ2+3Ψ02Ψ12,Ψ0ξ,ζ,ℑ=32ηsinh16ξ+ζ,Ψı+1ξ,ζ,ℑ=E−1sρE−∑ı=0∞ℬıΨξ−2∑ı=0∞ℬıΨξξξ−2∑ı=0∞ℬıΨξζζ,for ı=0,1,2,…,(41)Ψ1ξ,ζ,ℑ=E−1sρE−Ψ03ξ−2Ψ03ξξξ−2Ψ03ξζζ,Ψ1ξ,ζ,ℑ=−3η3sinh216ξ+ζcosh16ξ+ζ+38η3cosh316ξ+ζE−11sρ+2,Ψ1ξ,ζ,ℑ=−3η3sinh216ξ+ζcosh16ξ+ζ+38η3cosh316ξ+ζℑρΓρ+1.
The subsequent terms are(42)Ψ2ξ,ζ,ℑ=E−1sρE−3Ψ02Ψ1ξ−23Ψ02Ψ1ξξξ−23Ψ02Ψ1ξζζ,=332η5sinh16ξ+ζ765cosh416ξ+ζ−729cosh216ξ+ζ+91ℑ2ρΓ2ρ+1,Ψ3ξ,ζ,ℑ=E−1sρE−3Ψ02Ψ2+3Ψ02Ψ12ξ−23Ψ02Ψ2+3Ψ02Ψ12ξξξ−23Ψ02Ψ2+3Ψ02Ψ12ξζζ,=−3128cosh16ξ+ζ171738cosh616ξ+ζ−349884cosh416ξ+ζ+215496cosh216ξ+ζ−36907ℑ2ρΓ2ρ+1.
The EDM result is(43)Ψξ,ζ,ℑ=Ψ0ξ,ζ,ℑ+Ψ1ξ,ζ,ℑ+Ψ2ξ,ζ,ℑ+Ψ3ξ,ζ,ℑ+⋯,Ψξ,ζ,ℑ=32ηsinh16ξ+ζ−3η3sinh216ξ+ζcosh16ξ+ζ+38η3cosh316ξ+ζℑρΓρ+1+332η5sinh16ξ+ζ765cosh416ξ+ζ−729cosh216ξ+ζ+91ℑ2ρΓ2ρ+1−3128cosh16ξ+ζ171738cosh616ξ+ζ−349884cosh416ξ+ζ+215496cosh216ξ+ζ−36907ℑ2ρΓ2ρ+1+⋯.
Forρ=1,(44)Ψξ,ℑ=32ηsinh16ξ+ζ−ηℑ.
In Figure3, the exact and the EDM solutions of Example 2 at ρ=1 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 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 4, the first graph shows the two dimensions of exact and analytical solutions with respect to ℑ and the second one shows the different fractional-order graph with respect to ℑ.Figure 3
(a) The exact solution figure ofΨξ,ℑ of Example 2. (b) The EDM solution figure of Ψξ,ℑ of Example 2. (c) The EDM solution figure of Ψξ,ℑ at ρ=0.8 Example 2. (d) The EDM solution figure of Ψξ,ℑ at ρ=0.6 Example 2.
(a)(b)(c)(d)Figure 4
(a) The exact and EDM solution ofΨξ,ℑ with respect to ℑ Example 1. (b) The EDM solution figure of different fractional-order of ρ with respect to ℑ of Example 1.
(a)(b)


## 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.



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