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 ()-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 ()-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.

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 bywhere , , , , , , and . Additionally, is Planck’s constant; is the lower-hybrid resonance frequency; , are the ion (electron) gyrofrequency; is the ion mass; and 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, -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 ()-ZK equation with power-law nonlinearity that is given by [35–38].where , respectively, represent the nonlinearity and dispersion real valued constants. Also, is the evolution term while represents the power law nonlinearity parameter. Using the following wave transformation [39, 40] on Equation (1) where is an arbitrary constant yields

Integrating Equation (3) once with zero constant of the integration leads to

Through the balancing principle, the terms and force that . Thus, we employ another transformation on Equation (1) gives

Balancing between the terms of Equation (5) leads to

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 ()-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 ()-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 formwhere are arbitrary constants while is a function that satisfies the next ODEwhere are arbitrary constants. Exchanging the values of with Equation (6) along (7) and aggregation of all terms with the same power of then equating the gathering terms with zero lead to a system of equations. Solving this system yields

Family I

Family II

Family III

Family IV

Family VI

Thus, using the above families leads to the new exact solitary wave solutions to the Atangana conformable derivative ()-ZK equation with power-law nonlinearity in the next formulas.

For , we get

For , we get

For , we get

For , we get

For , we get