Abstract
In this article, we applied a new technique for solving the time-fractional coupled Korteweg-de Vries (KdV) equation. This method is a combination of the natural transform method with the Adomian decomposition method called the natural decomposition method (NDM). The solutions have been made in a convergent series form. To demonstrate the performances of the technique, two examples are provided.
1. Introduction
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 bysubject to initial conditionswhere ; , and 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.
2. Natural Transform
Definition 1. (see [8–11]). The natural transform of the function is defined bywhere and are the transform variables.
Definition 2. The inverse natural transform of is defined by
Now, we introduce some properties of the natural transform given as follows.
Property 3.
Property 4.
Theorem 5. (see [8–10]). If , where and is the natural transform of the function , then the natural transform of the Caputo fractional derivative is given by
3. 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), we get
Operating the inverse natural transform of equation (9), we obtain
The natural decomposition method represents the solution as infinite series
and the nonlinear terms decomposed aswhere , , and are Adomian polynomial, which can be calculated by
Substituting equations (11) and (12) into equation (10) yields
We get the general recursive formula
Finally, the approximate solutions can be written as
4. Applications
Now, we explain the appropriateness of the technique by the following examples.
Example 1. Consider the fractional coupled KdV equation (1) with , , , , and subject to
Solution 1. Applying the method formulated in Section 3 leads to the following:We define the following recursive formulas:This givesTherefore, the series solution can be written in the formFor , , and , 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 , , , , and 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].
Example 2. Consider the fractional coupled KdV in the form equation (1) with , , , , and subject to
Solution 2. Applying the method formulated in Section 3 leads to the following:The component of the solution given byThis givesTherefore, the series solution can be written in the formFor , the exact solution is
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(b)
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The approximate solutions (29) and (30) for the special cases are shown in Figures 3 and 4; the numerical results in Table 2 where show that the solutions obtained by the NDM are nearly identical with the exact solution. Also, it can be seen from Table 2 that the solutions obtained by the NDM are the same with those obtained in [26].
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(b)
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5. Conclusion
In this paper, the approximate solution of the time-fractional coupled KdV equation has been successfully done by using the NDM. We have tested the method on two examples, which revealed that the method is highly efficacious by comparing the approximate solutions with two different methods and exact solutions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.