Solution of Newell – Whitehead – Segal Equation of Fractional Order by Using Sumudu Decomposition Method

Newell Whitehead Segal (NWS) equation has been used in describing many natural phenomena arising in fluid mechanics and hence acquired more attention. Studies in the past gave importance to obtaining numerical or analytical solutions of this kind of equations by employing methods like Modified Homotopy Analysis Transform method (MHATM), Adomian Decomposition method (ADM), Homotopy Analysis Sumudu Transform method (HASTM), Fractional Complex Transform (FCT) coupled with He's polynomials method (FCT-HPM) and Fractional Residual Power Series method (FRPSM). This research aims to demonstrate an efficient analytical method called the Sumudu Decomposition Method (SDM) for the study of analytical and numerical solutions of the NWS of fractional order. The coupling of Adomian Decomposition method with Sumudu transform method simplifies the calculation. From the numerical results obtained, it is evident that SDM is easy to execute and offers accurate results for the NWS equation than with other methods such as FCT-HPM and FRPSM. Therefore, it is easy to apply the coupling of Adomian Decomposition technique with Sumudu transform method, and when applied to nonlinear differential equations of fractional order, it yields accurate results.

An NWS equation has been used in describing many natural phenomena arising in fluid mechanics and hence acquired more attention. This NWS equation describes the appearance of the stripe pattern in two-dimensional systems. Moreover, it has a lot of applications in fluid dynamics such as traveling wave patterns in binary fluids. Studies in the past gave importance to obtaining numerical or analytical solutions of this kind of equations by employing methods like Modified Homotopy Analysis Transform method (HASTM) [15], Adomian Decomposition method (ADM) [16], Homotopy Analysis Sumudu Transform method (HASTM) [17], Fractional Complex Transform (FCT) coupled with He's polynomials method (FCT-HPM) [18] and Fractional Residual Power Series method (FRPSM) [19].

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Solution of Newell -Whitehead -Segal Equation of Fractional Order by Using Sumudu Decomposition Method As shown previously, in this study, we applied the Sumudu decomposition method (SDM) to find an analytical and numerical solution to NWS equations of fractional order. SDM, simple and directly without any restrictive assumption as usual, is going in other methods. In many research papers SDM has also been applied to solve intricate problems in engineering, mathematics and applied science [20][21][22][23][24][25][26].
In the present research, we consider the fractional model of Newell-Whitehead-Segal (NWS) equation in the operator form: with initial condition:

Definition 4:
The Sumudu transform of  can be calculated as [14]:

Sumudu Decomposition Method (SDM)
In this section, we will briefly discuss SDM, to solve fractional-order nonlinear (1.1). By applying the Sumudu On employing the inverse ST for (3.2), we get Using (3.7), we define the following iterative formula [20]:

Elucidative Examples
In this section, we demonstrate the applicability of the previous method by giving examples.
Example1. By substituting with initial condition: a time-fractional linear NWS equation. By using SDM, (3.7) becomes )]].  ],  Fig. 1(a) and Fig. 1(b) respectively show the exact solution and approximate solutions yielded by the SDM at υ=1. Furthermore, the analysis of absolute errors is summarized in Table 1.
Therefore, the solution ( , ) x  in series form is given by;    The numerical results shown in Tables 2 and 3 and Figs.2 and 3 illustrate that SDM off ers accurate results in comparison with FCT-HPM [18] and FRPSM [19].

Conclusions
In this paper, SDM had been successfully applied to get approximate solutions of Newell-Whitehead-Segal (NWS) equation of fractional order. It is clearly seen from the numerical results that SDM is easy to execute and offers accurate results for the NWS equation. Hence, SDM is a simple and effective method to obtain approximate and analytical solutions for many differential fractional equations.