Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives

In this paper, we have studied the time-fractional Zakharov-Kuznetsov equation (TFZKE) via natural transform decomposition method (NTDM) with nonsingular kernel derivatives. The fractional derivative considered in Caputo-Fabrizio (CF) and Atangana-Baleanu derivative in Caputo sense (ABC). We employed natural transform (NT) on TFZKE followed by inverse natural transform, to obtain the solution of the equation. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results.


Introduction
Fractional calculus is an emerging field in various branches of engineering science. Fractional differential equations attracted researchers as they used to model a variety of diverse applications such as visco elasticity, heat conduction, biology, and dynamical systems [1][2][3][4][5][6][7]. Due to its importance in diverse fields, considerable methods developed to study the exact and computational solutions of fractional differential equations. Other than the modelling, divergence and convergence of the solutions are also equally important. A suitable definition is essential for a fractional generalization of a physical model. Several fractional derivative definitions developed in the last few decades. Some of the popular definitions in the literature are Riemann-Liouville (R-L), Caputo, CF, ABC, Grunwald-Letnikov, and Riesz fractional derivatives. For more details, we refer to [8,9] and the references therein. R-L and Caputo fractional derivatives have a singular kernel. Recently, two nonsingular kernel fractional derivative definitions are developed by Atangana-Baleanu and Caputo-Fabrizio. Several methods are being investigated for the analysis of fractional differential equations for accuracy and reliable solutions. Some of the popular semi analytical and numerical methods are variational iteration method (VIM) [10], fractional differential transform method [11][12][13][14], homotopy perturbation transform method (HPTM) [15], homotopy analysis transform method [16,17], residual power series method (RPS) [18], q-homotopy analysis transform method (q-HATM) [19][20][21], operational matrix method [22], tension spline method [23], parametric cubic spline [24], exponential B-spline method [25][26][27], fractional natural decomposition method (FNDM) [28], Adam Bashforth's Moulton method [29], and references therein.
In this paper, we have considered the TFZKE with initial condition This model illustrates the behavior of weakly nonlinear ion-acoustic waves in a plasma bearing cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This problem has been solved by many techniques such as VIM [10], homotopy perturbation method [30], HPTM [15], perturbation iteration algorithm and RPS [31], and new iterative Sumudu transform method [32]. Recently, Veeresha and Prakasha [28] presented the applications of q-HATM and FNDM for solving TFZKE.
The aim of this paper is to implement NTDM to solve TFZKE. Rawashdeh and Maitama [33] introduced NTDM for a class of nonlinear partial differential equations. NTDM do not require linearization; prescribe assumptions, perturbation, or discretization; and prevent any round-off errors. Recently, NTDM employed to time-fractional Fisher's equation [34] and ð2 + 1Þ − dimensional time-fractional coupled Burger equations [35]. The paper is organized as follows. Basic definitions of singular and nonsingular definitions of fractional calculus and NT and its fractional derivatives are discussed briefly in Section 2. In Section 4, we presented the convergence and uniqueness of the solutions. In Section 3, we presented the NTDM for nonsingular definitions to solve TFZKE. In Section 5, few examples of TFZKE are given to validate the present methods. Section 6 presents the results and discussions. In Section 7, brief conclusion of this paper is presented.

Basic Definitions
There are various fractional derivative definitions that are available in the literature; for more details, we refer [8,[36][37][38]. In this section, we give the definitions of R-L, Caputo, CF, and ABC fractional derivatives for the benefit of the readers.
Definition 4 (see [40]). The ABC fractional derivative of f ðωÞ is presented as where 0 < μ < 1. Normalization function is BðμÞ, and the Mittag-Leffler function is E μ ðzÞ = ∑ ∞ l=0 ðz l /Γðμl + 1ÞÞ. These definitions widely used to study the fractional differential equation solutions using numerous integral transform techniques such as Sumudu transform, Shehu transform, and Laplace transform. Recently, natural transform of these definitions applied to study various differential equations; for more details, we refer [33,41,42]. Computational time can For t ∈ ð0,∞Þ, natural transform of uðtÞ is defined by where HðtÞ is the Heaviside function.
Definition 6. The inverse natural transform of Uðs, vÞ is given by Lemma 7 (linearity property). If natural transform of u 1 ðtÞ is u 1 ðs, vÞ and u 2 ðtÞ is u 2 ðs, vÞ, then where c 1 and c 2 are constants.

Journal of Function Spaces
Definition 9 (see [43]). Natural transform of D μ t uðtÞ by means of Caputo sense is given as Definition 10 (see [44]). Natural transform of D μ t uðtÞ by means of CF is defined as With this motivation, we defined natural transform of ABC derivative as follows.
Definition 11. Natural transform of D μ t uðtÞ by means of ABC derivative is defined as

Methodology
In this section, we present a novel approximate analytical procedure based on natural transform [42] to the following equation with the initial condition where N , L, and hðζ, tÞ are nonlinear, linear, and source terms, respectively. Now we employing NT on equation (15) by considering fractional derivative by means of three fractional definitions.
Case 1. (NTDM CF ). By taking natural transform of equation (15), by means of CF fractional derivative, we obtain where By taking inverse natural transform using (8), we rewrite (17) as N ðuðζ, tÞÞ can be decomposed into where A l is the Adomian polynomials [45,46]. We assume that equation (15) has the analytical expansion By substituting equations (20) and (21) into (19), we obtain   Journal of Function Spaces From (22), we get By substituting (23) into (21), we get the NTDM CF solution of (15) and (16) as Case 2. (NTDM ABC ). By taking natural transform of equation (15), by means of ABC derivative, we acquire where N ðuðζ, tÞÞ can be decomposed into where A l is the Adomian polynomials. We assume that equation (15) has the analytical expansion By substituting equations (28) and (29) into (27), we obtain From (30), we get By substituting (31) into (29), we get the NTDM ABC solution of (15)-(16) as

Convergence Analysis
We have presented uniqueness and convergence of the NTDM CF and NTDM ABC in this section.

Numerical Results and Discussion
Tables 1 and 2 demonstrates the comparison of absolute errors with the existing methods and approximate solutions for different fractional orders with different fractional derivatives, respectively, of Example 1. Absolute errors of Example 1 graphically represented in Figure 1 for fixed t when μ = 1.
In Figure 2, we plotted approximate solutions for different values of μ for fixed t of Example 1. Figure 3 presents the comparison of NTDM CF and NTDM ABC solutions of Example 1 with exact solution for different values of fractional order μ for fixed x and ζ. In Table 3, we presented absolute errors of two fractional derivative solutions and existing results of Example 2. We have tabulated the approximate solution of Example 2 for noninteger fractional values in Table 4. For Example 2, NTDM CF and NTDM ABC absolute errors graphically represent in Figure 4 for fixed t when μ = 1, and in Figure 5, we plotted approximate solutions for different values of μ for specific value of t. Figure 6 presents the comparison of NTDM CF and NTDM ABC with exact solution for noninteger values of μ for fixed t and ζ of Example 2. It is observed from tables and figures that the two-term approximate solution is having good accordance with the existing results and exact solution. For noninteger values, NTDM CF and NTDM ABC are showing same behavior.

Conclusions
In this paper, we have studied the TFZKE through natural transformation by means of CF and ABC derivatives. We compared numerical results with the existing results. It is observed that the present method results are in accordance with existing methods. The NTDM is simple in its principles; also, NTDM is effective in solving nonlinear fractional differential equations, and promising method for a large varieties of such equations arises in mathematical physics.

Data Availability
There is no any data available.

Conflicts of Interest
The authors declare that they have no competing interests.