The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation

Newell-Whitehead-Segel (NWS) equation is a nonlinear partial differential equation used in modeling various phenomena arising in fluid mechanics. In recent years, various methods have been used to solve the NWS equation such as Adomian Decomposition method (ADM), Homotopy Perturbation method (HPM), New Iterative method (NIM), Laplace Adomian Decomposition method (LADM) and Reduced Differential Transform method (RDTM). In this study, the NWS equation is solved approximately using the Semi Analytical Iterative method (SAIM) to determine the accuracy and effectiveness of this method. Comparisons of the results obtained by SAIM with the exact solution and other existing results obtained by other methods such as ADM, LADM, NIM and RDTM reveal the accuracy and effectiveness of the method. The solution obtained by SAIM is close to the exact solution and the error function is close to zero compared to the other methods mentioned above. The results have been executed using Maple 17. For future use, SAIM is accurate, reliable, and easier in solving the nonlinear problems since this method is simple, straightforward, and derivative free and does not require calculating multiple integrals and demands less computational work.


Introduction
A NWS equation is a nonlinear Partial Differential equation (PDE) and it is used in modeling various phenomena arising in fluid mechanics. This equation is used for some problems in various systems, for example, Faraday instability, biological systems, nonlinear optics, Rayleigh-Benard convection, and chemical reactions. The equation acquired by Newell, Whitehead, and Segel is as follows [1-2]: (2) where , b c are real numbers and ,  m are positive integers.
In recent years, different methods have been used to solve the NWS equation. Ezzati and Shakibi [3] and Manaa [4] employ the Adomian Decomposition method (ADM) to obtain the numerical approximations of NWS. Saravanan and Magesh [5] conducted a comparative study in solving the NWS equation between Reduced Differential Transform Method (RDTM) and ADM. Nourazar et al. [6] and Jassim [7] solved the NWS equation using the Homotopy Perturbation method (HPM) and HPM using Laplace transform respectively. Also, Patade and Bhalekar [8] presented the application of the New Iterative method (NIM) to solve NWS equation. Prakash and Kumar in [9] presented the application of the Variational Iteration method (VIM) while Pue-on [10] applied the Laplace Adomian Decomposition method (LADM) to solve the NWS equation.
The Semi Analytical Iterative method (SAIM) was proposed by Temimi and Ansari to solve linear and nonlinear problems [11][12]. Al-Jawary et al. [13] solved the Fokker-Planck Equations (FPE) using SAIM to assemble the exact solution for the one dimension, two dimensions and three dimensions FPE. The method shows a very accurate and high order of convergence, reliable and effectual and also optional in any restrictive assumptions for non-linear terms. This can be supported in [14][15][16].
LADM was implemented to solve the approximate solution of nonlinear differential equations by Khuri [17]. Kiymaz [18] has used LADM to solve initial value problems. Fadaei applied LADM to the linear and nonlinear system of PDEs [19]. Khan et al. [20] proposed LADM in solving Nonlinear Coupled Partial Differential Equation.
This method was also successfully employed by Naghipour and Manafian [21] to solve the Burgers' equation.
There are several advantages of SAIM over the existing methods. SAIM is a very simple and easy method to be implemented. SAIM avoids the calculation of Adomian polynomials for a nonlinear term in ADM and thus demands less computational work. SAIM produces better approximate solutions among the other methods mentioned above.
The purpose of this paper is to employ SAIM to obtain the approximate solution to NWS equation. Illustrative examples in this paper will be solved numerically and the results obtained are compared with the exact solution and other existing results obtained by other methods such as LADM, NIM, RDTM and ADM to reveal the reliability and accuracy of the method.

Semi-Analytical Iterative Method (SAIM)
The basic idea of SAIM is presented in this section. Consider the general form of the differential equation as follows [16] where t is an independent variable,   , u x t is an unknown function,

 
, f x t is a known function, L is a linear operator, N is a nonlinear operator, and B is a boundary operator.
With SAIM, we first assume the initial problem as   0 , u x t and the work is as follows: Consider the next iteration for the solution as can be computed by solving the following equation This leads to the general equation of this method in the form as follows . This method is direct and straightforward. Continuing these steps will give a good approximate solution.

Applications
In this section, the Newell-Whitehead-Segel equation will be solved using SAIM. The Newell-Whitehead-Segel equation is given as below with the initial condition, b c are real numbers and ,  m are positive integers. By using SAIM, (6) will become Comparing (9) with (3) gives the following The initial problem with initial condition that needs to be solved is Thus we obtained For the second iterative, we need to solve the following Mathematics and Statistics 8(2): 87-94, 2020 The general form of iterative function to the solution that needs to be solved is given by

Illustrative Examples
Three numerical examples are considered to be solved numerically by SAIM in this section to reveal the reliability and accuracy of the method.

Example 1
Given the Newell-Whitehead-Segel equation as follows and the exact solution of this equation is given by By using SAIM, (19) becomes The initial problem with initial condition that needs to be solved is Thus, we obtained For the second iterative, we solve the following equation The next iterative to the solution can be obtained by solving Thus, with SAIM the first few iterative solutions are In this study, we let to execute the solution The results will be plotted in Figure 1.

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The Semi Analytics Iterative Method for Solving Newell-Whitehead-Segel Equation From Figure 2, at 0 t  , the accuracy of these three methods is similar since the magnitude of error is 0. However, as t increases, the result obviously shows that the magnitude errors of SAIM are lower than NIM and LADM. The error increases due to the range of utility of the power series which is limited to the neighborhood of the origin by its convergence radius that is determined by the singularity closest to that point. Based on Figure 3, at 0 t  there is no difference among these four methods but when t increases, obviously we can see that SAIM is in good agreement with the exact solution compared to NIM, ADM and RDTM.
From Figure 4, in the beginning all these three methods stayed constant at   , 0 u x t  . As t increases, SAIM shows better accuracy compared to NIM, ADM and RDTM. Thus, the result has shown that the accuracy of the magnitude value of SAIM is much better since the value of is close enough to zero compared to the other methods.

Conclusion
In this study, the Semi Analytical Iterative method (SAIM) is used to solve the Newell-Whitehead-Segel equation (NWS). As mentioned above, for the first example, the fifth iterative solution of the SAIM is compared with the exact solution, the four term iterative solution of the NIM and LADM in Figure 1  . It can be seen that, as time increases, SAIM shows better approximations than the others. A comparison of the magnitude error values between SAIM, Exact Solution, NIM, ADM and RDTM has been plotted in Figure 4. It shows that the relative error is consistently very small compared to the others. Again, this indicates the reliability, efficiency and accuracy of SAIM. The iteration of SAIM is direct and straightforward. SAIM is simple, accurate, reliable and useful in solving nonlinear problems. It is derivative free, does not require the calculation of multiple integrals and does not require the usage of Adomian polynomial.