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

Department of Mathematics, Faculty of Science, Jiangsu University, 212013 Zhenjiang, China Department of Mathematics, El Obour Institutes, 11828 Cairo, Egypt Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China Department of Basic Science, Higher Technological Institute 10th of Ramadan City, El Sharqia 44634, Egypt Department of Mathematics, Science Faculty, Firat University, 23119 Elazig, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan


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 loadvarying 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 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][13][14][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][17][18][19][20][21][22][23][24].
where a, b, respectively, represent the nonlinearity and dispersion real valued constants. Also, B t is the evolution term while n represents the power law nonlinearity parameter. Using the following wave transformation [39,40] ½B = Bðx , y, z, tÞ = PðFÞ, F = x + y + z + ðλ/αÞ ðt + ð1/ΓðαÞÞÞ α on Equation (1) where λ is an arbitrary constant yields λ P ′ + a n + 1 Integrating Equation (3) once with zero constant of the integration leads to λ P + a n + 1 Through the balancing principle, the terms P n+1 and P ″ force that m = 2/n. Thus, we employ another transformation P = U 2/n on Equation (1) gives λ U 2 + a n + 1 Balancing between the terms of Equation (5) leads to m = 1: The outline of this research paper is given as follows. Section 2 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.

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 where a 0 , a 1 , and b 1 are arbitrary constants while FðFÞ is a function that satisfies the next ODE where u, ρ, and δ are arbitrary constants. Exchanging the values of U, U″ with Equation (6) along (7) and aggregation of all terms with the same power of K j FðFÞ , ðj = −4,−3,⋯,3 , 4Þ then equating the gathering terms with zero lead to a system of equations. Solving this system yields Family I Family II Advances in Mathematical Physics Family III Family IV Family VI 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.

Numerical
Solutions. Here, we use three different analytical solutions Equations (16), (19) and (20) where c M , E M follow the next conditions, respectively: Substituting Equation (32) into Equation (5) gives ðM + 7Þ of equations. Resolving this system leads to the following values of exact, numerical, and absolute values or error.

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 is an arbitrary constants, λ is the frequency, and G is an arbitrary constant.

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 compar-ison between our obtained computational solutions and other previous obtained solutions, while the second axis of this discussion is studying our exact and numerical solutions.

Figure and Table Interpretation
This section illustrates our explained Figures 1-3 and Tables 1-3 with the abovementioned values of the parameters.
(i) Figure 1 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) Figure 2 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) Figure 3 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 solutions

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.  (16)

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this article.