# On the Analytical and Numerical Solutions in the Quantum Magnetoplasmas: The Atangana Conformable Derivative (1+3)-ZK Equation with Power-Law Nonlinearity

Authors: Mostafa M. A. Khater; Yu-Ming Chu; Raghda A. M. Attia; Mustafa Inc; Dianchen Lu
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this research paper, our work is connected with one of the most popular models in quantum magnetoplasma applications. The computational wave and numerical solutions of the Atangana conformable derivative (1+3)-Zakharov-Kuznetsov (ZK) equation with power-law nonlinearity are investigated via the modified Khater method and septic-B-spline scheme. This model is formulated and derived by employing the well-known reductive perturbation method. Applying the modified Khater (mK) method, septic B-spline scheme to the (1+3)-ZK equation with power-law nonlinearity after harnessing suitable wave transformation gives plentiful unprecedented ion-solitary wave solutions. Stability property is checked for our results to show their applicability for applying in the model’s applications. The result solutions are constructed along with their 2D, 3D, and contour graphical configurations for clarity and exactitude.


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## Body

## 1. Introduction

In the existence of a magnetized e-p-i plasma [1], the ZK equation is one of the widely common methods to characterize the ion-acoustic solitary waves. The magnetized load-varying dusty plasma is the best location to look for alternate placed dust ion acoustic waves of nonthermal electrons with a vortex-like spread of velocity [2]. In a comprehensive computational analysis, the ZK method was used to spread the dust-acoustic waves in a magnetized dusty plasma [3] and to excite the electrostatic ion-acoustic lone wave in two dimensions of negative ion magnetoplasmas of superthermal electrons [4]. This plasma comprises of nonthermal ions and negatively charged mobile dust crystals, and q-distributed temperature electrons of distinct nonextensivity power [5]. The ZK equation’s mathematical formula found by the well-known reductive disruption process [6] is given by(1)DtαB+LBBz+Bzzz+QBxxz+Byyz=0,where Dtα=dα/dtα,0<α<0,B=Bx,y,z,t, L=2, Q=1−He2/8, He=Hωciωce/2KBTFe, H=2π, ωci=eB0/2KBTFe, and ωce=eB0/me,c. Additionally, H is Planck’s constant; ωciωce is the lower-hybrid resonance frequency; ωci=eB0/Mic, ωce=eB0/Mec are the ion (electron) gyrofrequency; Mi is the ion mass; and c is the speed of light in vacuum.Solving this kind of models has attracted many researchers in various areas, chemical physics [7], geochemistry [8], plasma physics [8], fluid mechanics [9], optical fiber [10], solid-state physics [11], and so on [12–15]. Consequently, constructing the exact solutions of these mathematical models is an indispensable tool for detecting novel properties of them that can be used in their various applications. However, finding the exact solutions of them are not easy to process but is also considered a hard and complex process where there is no unified computational or numerical technique that is able to be applied to all nonlinear evolution (NLE) equations. Almost all computational and numerical techniques depend on an auxiliary equation that is considered a pivot tool in these techniques where all obtained solutions via these schemes are special cases of its general solutions [16–24].For the fractional models, many analytical and numerical methods with various fractional operators have been derived such as the exponential expansion method, Khater method, Kudryashov method, simplest equation method,Ψ′/Ψ-expansion method, Riccati expansion method, first integral method, tanh method, and the functional variable method [25–34].This paper studies the analytical and numerical solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity that is given by [35–38].(2)DtαB+aBnBx+bBxxx+Byyx+Bzzx=0,where a,b, respectively, represent the nonlinearity and dispersion real valued constants. Also, Bt is the evolution term while n represents the power law nonlinearity parameter. Using the following wave transformation [39, 40] B=Bx,y,z,t=PF,F=x+y+z+λ/αt+1/Γαα on Equation (1) where λ is an arbitrary constant yields(3)λP′+an+1PnP′+b3P″=0.Integrating Equation (3) once with zero constant of the integration leads to(4)λP+an+1Pn+1+3bP″=0.Through the balancing principle, the termsPn+1 and P″ force that m=2/n. Thus, we employ another transformation P=U2/n on Equation (1) gives(5)λU2+an+1U4+3b4−2nn2U′2+6bnUU″=0.Balancing between the terms of Equation (5) leads to m=1.The outline of this research paper is given as follows. Section2 employs the mK method and septic B-spline scheme to get the abundant explicit wave and numerical solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity. Section 3 investigates the stability of the results solutions. Section 4 shows and discusses the obtained results in our research paper. Section 5 gives the graphical demonstration of some of our solutions. Section 6 explains the conclusion of our study.


## 2. Implementation

In this section, we employ three recent analytical schemes to find the explicit wave solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity.


### 2.1. Ion-Acoustic Solitary Waves Solutions

This section gives a transitory elucidation of the mK method. We now explore a nontrivial solution for Equation (5) in the form(6)U=∑i=1maiKiFF+∑i=1mbiK−iFF+a0=a1KFF+a0+b1K−FF,where a0,a1,andb1 are arbitrary constants while FF is a function that satisfies the next ODE(7)F′F=ùK−FF+ρKFF+δlnK.where ù,ρ,andδ are arbitrary constants. Exchanging the values of U,U″ with Equation (6) along (7) and aggregation of all terms with the same power of KjFF,j=−4,−3,⋯,3,4 then equating the gathering terms with zero lead to a system of equations. Solving this system yieldsFamily I(8)a0⟶a1δ2−4ρù+a1δ2ρ,b1⟶0,λ⟶14−3bδ2−4bρù,a⟶−9bρ24a12,n⟶−4.Family II(9)a0⟶b1δ2−4ρù+b1δ2ù,a1⟶0,λ⟶14−3bδ2−4bρù,a⟶−9bù24b12,n⟶−4.Family III(10)a1⟶a0ρδ,b1⟶a0ùδ,λ⟶14−3bδ2−4bρù,a⟶−9bδ24a02,n⟶−4.Family IV(11)a0⟶a1δ2ρ,b1⟶0,λ⟶32bδ2−4bρù,a⟶−18bρ2a12,n⟶2.Family VI(12)a0⟶b1δ2ù,a1⟶0,λ⟶32bδ2−4bρù,a⟶−18bù2b12,n⟶2.Thus, using the above families leads to the new exact solitary wave solutions to the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity in the next formulas.Forδ2−4ρù<0,ρ≠0, we get(13)BI,1x,t=2a14ρù−δ2tan4ρù−δ24αH−3bϕδ2−4ρù/8α+δ2−4ρù/ρ,BI,2x,t=2a14ρù−δ2cot4ρù−δ24αH−3bϕδ2−4ρù/8α+δ2−4ρù/ρ,BII,1x,t=2b1δ2−4ρù+δ/ù−4ρ/δ−4ρù−δ2tan4ρù−δ24αH−3bϕδ2−4ρù/8α,BII,2x,t=2b1δ2−4ρù+δ/ù−4ρ/δ−4ρù−δ2cot4ρù−δ24αH−3bϕδ2−4ρù/8α,BIII,1x,t=1a0δ2−4ρù/δ−4ρù−δ2sin4ρù−δ2H−3bϕδ2−4ρù/4α+δcos4ρù−δ2H−3bϕδ2−4ρù/4α+δ,BIII,2x,t=1−a0δ2−4ρù/δ4ρù−δ2sin4ρù−δ2H−3bϕδ2−4ρù/4α+δcos4ρù−δ2H−3bϕδ2−4ρù/4α−δ,BIV,1x,t=a14ρù−δ2tan4ρù−δ23bϕδ2−4ρù+2αH/4α2ρ,BIV,2x,t=a14ρù−δ2cot4ρù−δ23bϕδ2−4ρù+2αH/4α2ρ,BV,1x,t=12b1δù−4ρδ−4ρù−δ2tan4ρù−δ23bϕδ2−4ρù+2αH/4α,BV,2x,t=12b1δù−4ρδ−4ρù−δ2cot4ρù−δ23bϕδ2−4ρù+2αH/4α.Forρδ2−4ρù<0,≠0, we get(14)BI,3x,t=2−a1δ2−4ρùtanhδ2−4ρù4αH−3bϕδ2−4ρù/8α−1/ρ,BI,4x,t=2−a1δ2−4ρùcothδ2−4ρù4αH−3bϕδ2−4ρù/8α−1/ρ,BII,3x,t=2b1δ2−4ρù+δ/ù−4ρ/δ2−4ρùtanhδ2−4ρù4αH−3bϕδ2−4ρù/8α+δ,BII,4x,t=2b1δ2−4ρù+δ/ù−4ρ/δ2−4ρùcothδ2−4ρù4αH−3bϕδ2−4ρù/8α+δ,BIII,3x,t=1a0δ2−4ρù/δδ2−4ρùsinhδ2−4ρùH−3bϕδ2−4ρù/4α+δcoshδ2−4ρùH−3bϕδ2−4ρù/4α+δ,BIII,4x,t=1−a0δ2−4ρù/δδ2−4ρùsinhδ2−4ρùH−3bϕδ2−4ρù/4α+δcoshδ2−4ρùH−3bϕδ2−4ρù/4α−δ,BIV,3x,t=−a1δ2−4ρùtanhδ2−4ρù3bϕδ2−4ρù+2αH/4α2ρ,BIV,4x,t=−a1δ2−4ρùcothδ2−4ρù3bϕδ2−4ρù+2αH/4α2ρ,BV,3x,t=12b1δù−4ρδ2−4ρùtanhδ2−4ρù3bϕδ2−4ρù+2αH/4α+δ,BV,4x,t=12b1δù−4ρδ2−4ρùcothδ2−4ρù3bϕδ2−4ρù+2αH/4α+δ.Forρù>0,ù≠0,ρ≠0,δ=0, we get(15)BI,5x,t=1a1ρùtanρù3bρùϕ/α+H+ρ−ù/ρ,BI,6x,t=1a1ρ−ù−ρùcotρù3bρùϕ/α+H/ρ,BII,5x,t=1b1ρùcotρù3bρùϕ/α+H+ρ−ù/ù,BII,6x,t=1b1ρ−ù−ρùtanρù3bρùϕ/α+H/ù,BIV,5x,t=a1ùtanρùH−6bρùϕ/αρù,BIV,6x,t=−a1ùcotρùH−6bρùϕ/αρù,BV,5x,t=b1ρcotρùH−6bρùϕ/αρù,BV,6x,t=−b1ρtanρùH−6bρùϕ/αρù.Forρù<0,ù≠0,ρ≠0,δ=0, we get(16)BI,7x,t=1a1ùtanhρ−ù3bρùϕ/α+H−1/ρ−ù,(17)BI,8x,t=1a1ùcothρ−ù3bρùϕ/α+H−1/ρ−ù,(18)BII,7x,t=1b1ρ−ùcothρ−ù3bρùϕ/α+H+1/ù,(19)BII,8x,t=1b1ρ−ùtanhρ−ù3bρùϕ/α+H+1/ù,(20)BIV,7x,t=a1ùtanhρ−ùH−6bρùϕ/αρ−ù,(21)BIV,8x,t=a1ùcothρ−ùH−6bρùϕ/αρ−ù,(22)BV,7x,t=b1ρcotρùH−6bρùϕ/αù,(23)BV,8x,t=b1ρ−ùtanhρ−ùH−H−6bρùϕ/αù.Forδ=ù/2=κ,ρ=0, we get(24)BII,9x,t=2b14/eκH−3bκ3ϕ/4α−2+κ/κ2+1,BIII,9x,t=1a02/eκH−3bκ3ϕ/4α−2+1,BVI,9x,t=14b14e3bκ3ϕ/2α+κH−2+1.Forδ=ρ=κ,ù=0, we get(25)BI,10x,t=2a1κ/κ2−cothκH/2−3bκ3ϕ/8α,BIII,10x,t=1−a0/eκH−3bκ3ϕ/4α−1,BIV,10x,t=−12a1coth3bκ3ϕ4α+κH2.Forù=0,δ≠0,ρ≠0, we get(26)BI,11x,t=2a1δ−4/ρeδH−3bδ3ϕ/4α−2−1+δ2/ρ,BIII,10x,t=12−a0/ρeδH−3bδ∧33ϕ/4α−2,BIV,10x,t=a1δ4/2−ρe3bδ∧3ϕ/2α+δH−12ρ.Forδ=ρ=0,ù≠0, we get(27)BII,10x,t=1b1/λtù+xù,BV,10x,t=b1λtù+xù.Forδ=0,ù=ρ, we get(28)BI,11x,t=1a1tan3bù3ϕ/α+C+Hù+−ù/ù,BII,11x,t=1b1cot3bù3ϕ/α+C+Hù+−ù/ù,BIV,11x,t=1b1cot3bù3ϕ/α+C+Hù+−ù/ù,BV,11x,t=a1tan−6bù3ϕα+C+Hù,BVI,11x,t=b1cot−6bù3ϕα+C+Hù.Forρ=0,δ≠0,ù≠0, we get(29)BII,12x,t=12b12δδeδH−3bδ3ϕ/4α−ù+δ2+δù,BIII,12x,t=a0ùδeδH−3bδ3ϕ/4α−ù+1,BVI,12x,t=12b1δ2δe3bδ3ϕ/2α+δH−ù+1ù.whereH=x+y+z,ϕ=t+1/Γαα.


### 2.2. Numerical Solutions

Here, we use three different analytical solutions Equations (16), (19) and (20) to evaluate the numerical solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity. Employing the septic spline technique to Equation (5) with the following conditions a1=3,a0=−6,a=−5/4,b=5,b1=0,δ=0,λ=−60,n=−4,ρ=−1,ù=4&a0=−25,a1=0,a=−9/20,b1=5,b=2,δ=0,λ=−150,n=−4,ρ=25,ù=−1&a1=3,a0=0,a=−4,b=2,b1=0,δ=0,λ=108,n=2,ρ=−1,andù=9 gives its numerical solutions in the next form(30)BF=∑T=−1M+1CMEM,where cM,EM follow the next conditions, respectively:(31)LBF=FFM,BFMwhereM=0,1,⋯,nETF=1S5F−FT−47,F∈FT−4,FT−3,F−FT−47−8F−FT−37,F∈FT−3,FT−2,F−FT−47−8F−FT−37+28F−FT−27,F∈FT−2,FT−1,F−FT−47−8F−FT−37+28F−FT−27+56F−FT−17,F∈FT−1,FT,FT+4−F7−8FT+3−F7+28FT+2−F7+56FT+1−F7,F∈FT,FT+1,FT+4−F7−8FT+3−F7+28FT+2−F7,F∈FT+1,FT+2,FT+4−F7−8FT+3−F7,F∈FT+2,FT+3,FT+4−F7,F∈FT+3,FT+4,0,otherwise.ForT∈−3,M+3, we get(32)BTF=CT−3+120CT−2+1191CT−1+2416CT+1191CT+1+120CT+2+CT+3.Substituting Equation (32) into Equation (5) gives M+7 of equations. Resolving this system leads to the following values of exact, numerical, and absolute values or error.


## 2.1. Ion-Acoustic Solitary Waves Solutions

This section gives a transitory elucidation of the mK method. We now explore a nontrivial solution for Equation (5) in the form(6)U=∑i=1maiKiFF+∑i=1mbiK−iFF+a0=a1KFF+a0+b1K−FF,where a0,a1,andb1 are arbitrary constants while FF is a function that satisfies the next ODE(7)F′F=ùK−FF+ρKFF+δlnK.where ù,ρ,andδ are arbitrary constants. Exchanging the values of U,U″ with Equation (6) along (7) and aggregation of all terms with the same power of KjFF,j=−4,−3,⋯,3,4 then equating the gathering terms with zero lead to a system of equations. Solving this system yieldsFamily I(8)a0⟶a1δ2−4ρù+a1δ2ρ,b1⟶0,λ⟶14−3bδ2−4bρù,a⟶−9bρ24a12,n⟶−4.Family II(9)a0⟶b1δ2−4ρù+b1δ2ù,a1⟶0,λ⟶14−3bδ2−4bρù,a⟶−9bù24b12,n⟶−4.Family III(10)a1⟶a0ρδ,b1⟶a0ùδ,λ⟶14−3bδ2−4bρù,a⟶−9bδ24a02,n⟶−4.Family IV(11)a0⟶a1δ2ρ,b1⟶0,λ⟶32bδ2−4bρù,a⟶−18bρ2a12,n⟶2.Family VI(12)a0⟶b1δ2ù,a1⟶0,λ⟶32bδ2−4bρù,a⟶−18bù2b12,n⟶2.Thus, using the above families leads to the new exact solitary wave solutions to the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity in the next formulas.Forδ2−4ρù<0,ρ≠0, we get(13)BI,1x,t=2a14ρù−δ2tan4ρù−δ24αH−3bϕδ2−4ρù/8α+δ2−4ρù/ρ,BI,2x,t=2a14ρù−δ2cot4ρù−δ24αH−3bϕδ2−4ρù/8α+δ2−4ρù/ρ,BII,1x,t=2b1δ2−4ρù+δ/ù−4ρ/δ−4ρù−δ2tan4ρù−δ24αH−3bϕδ2−4ρù/8α,BII,2x,t=2b1δ2−4ρù+δ/ù−4ρ/δ−4ρù−δ2cot4ρù−δ24αH−3bϕδ2−4ρù/8α,BIII,1x,t=1a0δ2−4ρù/δ−4ρù−δ2sin4ρù−δ2H−3bϕδ2−4ρù/4α+δcos4ρù−δ2H−3bϕδ2−4ρù/4α+δ,BIII,2x,t=1−a0δ2−4ρù/δ4ρù−δ2sin4ρù−δ2H−3bϕδ2−4ρù/4α+δcos4ρù−δ2H−3bϕδ2−4ρù/4α−δ,BIV,1x,t=a14ρù−δ2tan4ρù−δ23bϕδ2−4ρù+2αH/4α2ρ,BIV,2x,t=a14ρù−δ2cot4ρù−δ23bϕδ2−4ρù+2αH/4α2ρ,BV,1x,t=12b1δù−4ρδ−4ρù−δ2tan4ρù−δ23bϕδ2−4ρù+2αH/4α,BV,2x,t=12b1δù−4ρδ−4ρù−δ2cot4ρù−δ23bϕδ2−4ρù+2αH/4α.Forρδ2−4ρù<0,≠0, we get(14)BI,3x,t=2−a1δ2−4ρùtanhδ2−4ρù4αH−3bϕδ2−4ρù/8α−1/ρ,BI,4x,t=2−a1δ2−4ρùcothδ2−4ρù4αH−3bϕδ2−4ρù/8α−1/ρ,BII,3x,t=2b1δ2−4ρù+δ/ù−4ρ/δ2−4ρùtanhδ2−4ρù4αH−3bϕδ2−4ρù/8α+δ,BII,4x,t=2b1δ2−4ρù+δ/ù−4ρ/δ2−4ρùcothδ2−4ρù4αH−3bϕδ2−4ρù/8α+δ,BIII,3x,t=1a0δ2−4ρù/δδ2−4ρùsinhδ2−4ρùH−3bϕδ2−4ρù/4α+δcoshδ2−4ρùH−3bϕδ2−4ρù/4α+δ,BIII,4x,t=1−a0δ2−4ρù/δδ2−4ρùsinhδ2−4ρùH−3bϕδ2−4ρù/4α+δcoshδ2−4ρùH−3bϕδ2−4ρù/4α−δ,BIV,3x,t=−a1δ2−4ρùtanhδ2−4ρù3bϕδ2−4ρù+2αH/4α2ρ,BIV,4x,t=−a1δ2−4ρùcothδ2−4ρù3bϕδ2−4ρù+2αH/4α2ρ,BV,3x,t=12b1δù−4ρδ2−4ρùtanhδ2−4ρù3bϕδ2−4ρù+2αH/4α+δ,BV,4x,t=12b1δù−4ρδ2−4ρùcothδ2−4ρù3bϕδ2−4ρù+2αH/4α+δ.Forρù>0,ù≠0,ρ≠0,δ=0, we get(15)BI,5x,t=1a1ρùtanρù3bρùϕ/α+H+ρ−ù/ρ,BI,6x,t=1a1ρ−ù−ρùcotρù3bρùϕ/α+H/ρ,BII,5x,t=1b1ρùcotρù3bρùϕ/α+H+ρ−ù/ù,BII,6x,t=1b1ρ−ù−ρùtanρù3bρùϕ/α+H/ù,BIV,5x,t=a1ùtanρùH−6bρùϕ/αρù,BIV,6x,t=−a1ùcotρùH−6bρùϕ/αρù,BV,5x,t=b1ρcotρùH−6bρùϕ/αρù,BV,6x,t=−b1ρtanρùH−6bρùϕ/αρù.Forρù<0,ù≠0,ρ≠0,δ=0, we get(16)BI,7x,t=1a1ùtanhρ−ù3bρùϕ/α+H−1/ρ−ù,(17)BI,8x,t=1a1ùcothρ−ù3bρùϕ/α+H−1/ρ−ù,(18)BII,7x,t=1b1ρ−ùcothρ−ù3bρùϕ/α+H+1/ù,(19)BII,8x,t=1b1ρ−ùtanhρ−ù3bρùϕ/α+H+1/ù,(20)BIV,7x,t=a1ùtanhρ−ùH−6bρùϕ/αρ−ù,(21)BIV,8x,t=a1ùcothρ−ùH−6bρùϕ/αρ−ù,(22)BV,7x,t=b1ρcotρùH−6bρùϕ/αù,(23)BV,8x,t=b1ρ−ùtanhρ−ùH−H−6bρùϕ/αù.Forδ=ù/2=κ,ρ=0, we get(24)BII,9x,t=2b14/eκH−3bκ3ϕ/4α−2+κ/κ2+1,BIII,9x,t=1a02/eκH−3bκ3ϕ/4α−2+1,BVI,9x,t=14b14e3bκ3ϕ/2α+κH−2+1.Forδ=ρ=κ,ù=0, we get(25)BI,10x,t=2a1κ/κ2−cothκH/2−3bκ3ϕ/8α,BIII,10x,t=1−a0/eκH−3bκ3ϕ/4α−1,BIV,10x,t=−12a1coth3bκ3ϕ4α+κH2.Forù=0,δ≠0,ρ≠0, we get(26)BI,11x,t=2a1δ−4/ρeδH−3bδ3ϕ/4α−2−1+δ2/ρ,BIII,10x,t=12−a0/ρeδH−3bδ∧33ϕ/4α−2,BIV,10x,t=a1δ4/2−ρe3bδ∧3ϕ/2α+δH−12ρ.Forδ=ρ=0,ù≠0, we get(27)BII,10x,t=1b1/λtù+xù,BV,10x,t=b1λtù+xù.Forδ=0,ù=ρ, we get(28)BI,11x,t=1a1tan3bù3ϕ/α+C+Hù+−ù/ù,BII,11x,t=1b1cot3bù3ϕ/α+C+Hù+−ù/ù,BIV,11x,t=1b1cot3bù3ϕ/α+C+Hù+−ù/ù,BV,11x,t=a1tan−6bù3ϕα+C+Hù,BVI,11x,t=b1cot−6bù3ϕα+C+Hù.Forρ=0,δ≠0,ù≠0, we get(29)BII,12x,t=12b12δδeδH−3bδ3ϕ/4α−ù+δ2+δù,BIII,12x,t=a0ùδeδH−3bδ3ϕ/4α−ù+1,BVI,12x,t=12b1δ2δe3bδ3ϕ/2α+δH−ù+1ù.whereH=x+y+z,ϕ=t+1/Γαα.


## 2.2. Numerical Solutions

Here, we use three different analytical solutions Equations (16), (19) and (20) to evaluate the numerical solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity. Employing the septic spline technique to Equation (5) with the following conditions a1=3,a0=−6,a=−5/4,b=5,b1=0,δ=0,λ=−60,n=−4,ρ=−1,ù=4&a0=−25,a1=0,a=−9/20,b1=5,b=2,δ=0,λ=−150,n=−4,ρ=25,ù=−1&a1=3,a0=0,a=−4,b=2,b1=0,δ=0,λ=108,n=2,ρ=−1,andù=9 gives its numerical solutions in the next form(30)BF=∑T=−1M+1CMEM,where cM,EM follow the next conditions, respectively:(31)LBF=FFM,BFMwhereM=0,1,⋯,nETF=1S5F−FT−47,F∈FT−4,FT−3,F−FT−47−8F−FT−37,F∈FT−3,FT−2,F−FT−47−8F−FT−37+28F−FT−27,F∈FT−2,FT−1,F−FT−47−8F−FT−37+28F−FT−27+56F−FT−17,F∈FT−1,FT,FT+4−F7−8FT+3−F7+28FT+2−F7+56FT+1−F7,F∈FT,FT+1,FT+4−F7−8FT+3−F7+28FT+2−F7,F∈FT+1,FT+2,FT+4−F7−8FT+3−F7,F∈FT+2,FT+3,FT+4−F7,F∈FT+3,FT+4,0,otherwise.ForT∈−3,M+3, we get(32)BTF=CT−3+120CT−2+1191CT−1+2416CT+1191CT+1+120CT+2+CT+3.Substituting Equation (32) into Equation (5) gives M+7 of equations. Resolving this system leads to the following values of exact, numerical, and absolute values or error.


## 3. Stability Characteristics

In this section, the stability property has been tested of the obtained results based on the Hamiltonian system characteristics. This system imposes a single condition to ensure the stability of the solution. This condition is given by(33)∂M∂λλ=G>0,where M=1/2∫−EEB2dF where E is an arbitrary constants, λ is the frequency, and G is an arbitrary constant.Applying the stability check of Equation (20) with the following values of the parameters a1=3,a0=0,a=−4,b=2,b1=0,δ=0,λ=108,n=2,ρ=−1,andù=9, leads to(34)∂M∂λλ=6=−2.3447910280083306×10−13<0.Consequently, this solution is not stable and applying the same steps to other obtained solutions investigates their stability property.


## 4. Result and Discussion

Here, we discuss our obtained solutions of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity that have been obtained through one of the most recent computational schemes in nonlinear evolution equation field (the mK method) via two main axes which are a comparison between our obtained computational solutions and other previous obtained solutions, while the second axis of this discussion is studying our exact and numerical solutions.(i)
Computational solutions(1)
Applying the modified Khater method to the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity has obtained sixty distinct traveling wave solutions(2)
The difference between our obtained solutions and that have been obtained in [41] by Aminikhah et al. who had used the functional variable method; however, they have just found three solutions and accurate in their and our solutions, we can figure out the complete difference between these solutions that thing makes our solutions are novel(ii)
Numerical solutions(1)
Applying the septic B-spline scheme to the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity by using three of our obtained solutions in evaluating the initial and boundary conditions that give the ability of employing the septic B-spline scheme to the fractional model


## 5. Figure and Table Interpretation

This section illustrates our explained Figures1–3 and Tables 1–3 with the abovementioned values of the parameters.(i)
Figure1 and Table 1 show the value of the exact and numerical solutions and absolute error of Equation (5) with Equation (16) in three distinct types of sketches to explain the convergence between the two types of solutions(ii)
Figure2 and Table 2 show the value of exact and numerical solutions and absolute error of Equation (5) with Equation (19) in three distinct types of sketches to illustrate the closer between the two types of solutions(iii)
Figure3 and Table 3 explain the value of exact, numerical solutions and absolute error of Equation (5) with Equation (20) in three distinct types of sketches to show the matching between the two types of solutionsFigure 1
Exact, and numerical solutions based on the obtained analytical solution Equation (16) and septic B-spline scheme.Figure 2
Exact and numerical solutions based on the obtained analytical solution Equation (19) and septic B-spline scheme.Figure 3
Exact and numerical solutions based on the obtained analytical solution Equation (20) and septic B-spline scheme.Table 1
Exact and numerical value of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity through the obtained analytical solutions via the modified Khater method Equation (16) and septic B-spline scheme.
Value ofFExactNumericalAbsolute error00. -0.408248 I0. -0.0000110311 I0.4082370.00010. -0.408289 I0. -8.37587×10−6I0.4082810.00020. -0.40833 I0. -5.50221×10−6I0.4083240.00030. -0.408371 I0. -2.53611×10−6 I0.4083680.00040. -0.408412 I0. -1.13958×10−6 I0.4084110.00050. -0.408453 I0. +1.58999×10−7 I0.4084530.00060. -0.408493 I0. -1.01297×10−6 I0.4084920.00070. -0.408534 I0. -2.41857×10−6 I0.4085320.00080. -0.408575 I0. -5.26488×10−6 I0.408570.00090. -0.408616 I0. -7.96518×10−6 I0.4086080.0010. -0.408657 I0. -0.0000103138 I0.408647Table 2
Exact and numerical value of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity through the obtained analytical solutions via the modified Khater method Equation (19) and septic B-spline scheme.
Value ofFExactNumericalAbsolute error00. -0.2 I0. -0.0000794823 I0.1999210.00010. -0.19995 I0. -0.0000595992 I0.199890.00020. -0.1999 I0. -0.0000399605 I0.199860.00030. -0.19985 I0. -0.0000214 I0.1998290.00040. -0.1998 I0. -8.76169×10−6 I0.1997920.00050. -0.19975 I0. +1.47924×10−6 I0.1997520.00060. -0.199701 I0. -7.5007×10−6 I0.1996930.00070. -0.199651 I0. -0.0000199929 I0.1996310.00080. -0.199601 I0. -0.0000376969 I0.1995640.00090. -0.199552 I0. -0.000056177 I0.1994950.0010. -0.199502 I0. -0.0000736582 I0.199428Table 3
Exact and numerical value of the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity through the obtained analytical solutions via the modified Khater method Equation (20) and septic B-spline scheme.
Value ofFExactNumericalAbsolute error003.46945×10−173.46945×10−170.00010.00270.0007792360.001920760.00020.0054-8.67362E-180.00540.00030.00810.004895440.003204550.00040.010800.01080.00050.01356.50521E-190.01350.00060.016200.01620.00070.0189-4.33681E-190.01890.00080.0216-8.67362E-190.02160.00090.02429990.01415650.01014350.0010.02699990.02699993.46945×10−18


## 6. Conclusion

This paper has succeeded in the implementation of the mK method and septic B-spline scheme to the Atangana conformable derivative (1+3)-ZK equation with power-law nonlinearity. Sixty distinct novel computational solutions have been obtained. Three of these solutions have been used to evaluate the initial and boundary conditions that have allowed the application of the numerical scheme. Calculating the absolute value of error between the exact and numerical is the aim of our study. Moreover, the stability of our obtained solutions has been illustrated based on the Hamiltonian system characteristics. The effectiveness and power of our two used schemes have been verified, and all obtained solutions have been also verified by putting them back in the original equation via Mathematica 12 software.



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