A New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Method

In the latest years, the branch fractional calculus [1–5] has played a significant role in applied mathematics; this is evident when it is used in phenomena of physical science and engineering, which are described by fractional differential equations. Fractional partial differential equations (FPDEs) have lately been studied and solved in several ways [6, 7]. Many transforms coupled with other techniques were used to solve differential equations [8–10]. The coupled natural transform [11–14] and Adomian decomposition method [15–17] called the natural decomposition method (NDM) is introduced in [18, 19] to solve differential equations, and it presents the approximate solution in the series form. The natural decomposition method has been used by many researchers to find approximate analytical solutions; it has shown reliable and closely converged results of the solution. In [20], Eltayeb et al. used the NDM to take out an analytical solution of fractional telegraph equation. Khan et al. in [21] obtained the solution of fractional heat and wave equations by NDM. Rawashdeh and Al-Jammal [22] gave the solution of fractional ODEs using the NDM, and in [23], Shah et al. obtained the solution of fractional partial differential equations with proportional delay by using the NDM. Many analytical and numerical methods were used to solve the fractional coupled KdV equation, such as spectral collection method [24], HPM [25], DTM [26], VIM [27], and meshless spectral method [28]. In this paper, we provided the application of the NDM to find the approximate solutions of nonlinear time-fractional coupled KdV equations given by


Introduction
In the latest years, the branch fractional calculus [1][2][3][4][5] has played a significant role in applied mathematics; this is evident when it is used in phenomena of physical science and engineering, which are described by fractional differential equations. Fractional partial differential equations (FPDEs) have lately been studied and solved in several ways [6,7]. Many transforms coupled with other techniques were used to solve differential equations [8][9][10]. The coupled natural transform [11][12][13][14] and Adomian decomposition method [15][16][17] called the natural decomposition method (NDM) is introduced in [18,19] to solve differential equations, and it presents the approximate solution in the series form. The natural decomposition method has been used by many researchers to find approximate analytical solutions; it has shown reliable and closely converged results of the solution.
In [20], Eltayeb et al. used the NDM to take out an analytical solution of fractional telegraph equation. Khan et al. in [21] obtained the solution of fractional heat and wave equations by NDM. Rawashdeh and Al-Jammal [22] gave the solution of fractional ODEs using the NDM, and in [23], Shah et al. obtained the solution of fractional partial differential equations with proportional delay by using the NDM. Many ana-lytical and numerical methods were used to solve the fractional coupled KdV equation, such as spectral collection method [24], HPM [25], DTM [26], VIM [27], and meshless spectral method [28]. In this paper, we provided the application of the NDM to find the approximate solutions of nonlinear time-fractional coupled KdV equations given by subject to initial conditions where η, λ < 0; γ,μ, and ν are constant parameters. The KdV equation arose in many physical phenomena and application in the study of shallow-water waves, and it has been studied by many researchers. This work is organized as follows. In Section 2, we give definitions and properties of the natural transform. In Section 3, the NDM is made. Section 4 discusses the new technique and compares it with two different techniques by two examples and presents tables and graphs to offer the validation of the NDM. Discussion and conclusion are included.

Natural Transform
Definition 1. (see [8][9][10][11]). The natural transform of the function yðtÞ is defined by where u and s are the transform variables.
Definition 2. The inverse natural transform of Gðs, uÞ is defined by Now, we introduce some properties of the natural transform given as follows.

Analysis of Method
We explain the algorithm of the NDM by considering the fractional coupled KdV equations (1) and (2).
Applying natural transform to equation (1), we have Substituting the initial conditions of equation (2) into equation (8) Operating the inverse natural transform of equation (9), we obtain The natural decomposition method represents the solution as infinite series Abstract and Applied Analysis and the nonlinear terms decomposed as where A n , B n , and C n are Adomian polynomial, which can be calculated by Substituting equations (11) and (12) into equation (10) We get the general recursive formula Finally, the approximate solutions can be written as

Applications
Now, we explain the appropriateness of the technique by the following examples.

Abstract and Applied Analysis
This gives   Abstract and Applied Analysis Therefore, the series solution can be written in the form For α = β = 1, b = 3, and a = r, the exact solution of example (1) is The approximate solutions (22) and (23) for the special cases are shown in Figures 1 and 2; the numerical results in Table 1 where α = β = 1, a = 1, b = 3, ζ = 1, and r = 1 show that the solutions obtained by the NDM are nearly identical with the exact solution. Also, it can be seen from Table 1 that the solutions obtained by the NDM are more accurate than those obtained in [27].
Solution 2. Applying the method formulated in Section 3 leads to the following: