HE'S Variational Iteration Method for the Solution of Nonlinear Newell-Whitehead-Segel Equation

: In this paper, we apply He's Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three different cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an effective and highly promising method for solving different cases of nonlinear Newell-Whitehead-Segel equation. Abstract In this paper, we apply He’s Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three diﬀerent cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an eﬀective and highly promising method for solving diﬀerent cases of nonlinear Newell-Whitehead-Segel equation.


Introduction
We have to face many real-life time mathematical models for semi-analytical solution of nonlinear differential equations in day-today life. We also know that most of the nonlinear differential equations do not have an analytical solution. But by using He's variational iteration method (VIM), we can solve nonlinear differential equations, which was first envisioned by Professor Ji-He for solving a wide range of problems whose mathematical model yields differential equation or system of differential equations [2][3][4][5][6][7]. Later on He's variational iteration method (VIM) which is a semi-analytical method is applied to solve the nonlinear non-homogeneous partial differential equations [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Newell-Whitehead-Segel equation is solved by using the Adomian decomposition and multi-quadric quasi-interpolation methods [1] and Homotopy perturbation method [20].
In this article, we apply He's variational iteration method to obtain the solution of the non-linear Newell-Whitehead-Segel equation. Three different cases of nonlinear Newell-Whitehead-Segel equations are solved by using this method. The Newell-Whitehead-Segel equation models the interaction of the effect of the diffusion term with the nonlinear effect of the reaction term. The Newell-Whitehead-Segel equation is written as where a, b and k are real numbers with k > 0 and q is a positive integer. In equation (1.1) the first term on the left hand side, ∂u ∂t , expresses the variations of u(x, t) with time at a fixed location, the first term on the right hand side, ∂ 2 u ∂x 2 , expresses the variations of u(x, t) with spatial variable x at a specific time and the remaining terms on the right hand side au − bu q , takes into account the effect of the source term. In equation (1.1), u(x, t) is a function of the spatial variable x and the temporal variable t with x ∈ R and t ≥ 0. The function u(x, t) may be thought of as the (nonlinear) distribution of temperature in an infinitely thin and long rod or as the flow velocity of a fluid in an infinitely long pipe with small diameter. The Newell-Whitehead-Segel equations have wide applicability in mechanical and chemical engineering, ecology, biology and bio-engineering.
2. Proposed He's Variational iteration method for the Newell-Whitehead-Segel equation Consider the differential equation where L and N are Linear and nonlinear operators respectively and g(x) is inhomogeneous function. J. H. He suggested the variational iteration method where a correction functional for equation (2.1) can be written as where λ is Lagrange's multiplier, which can be identified optimally by the variational theory andũ n (t) as restricted variation that means δũ n = 0. It is to be noted that Lagrange multiplier λ can be taken as a constant or function. There are two steps in variational iteration method, first to find Lagrange's multiplier that can be find out optimally via integration by parts and by using the restricted variation, should be used for the determination of the successive approximations u n+1 (x), n ≥ 0 of the solution u(x). The zeroth approximation u 0 can be any selective function or can be any other initial condition that can be used at the initial stage or using the initial values u(0), u (0) and u (0) are preferably used for the selective zeroth approximation u 0 as will be seen later. Finally the solution is given by u(x) = lim n→∞ u n (x). Then u(x) will be the solution of given differential equation.

The Newell-Whitehead-Segel equation
To illustrate the capability and reliability of this method, three different cases of nonlinear Newell-Whitehead-Segel equation are presented.

Case I.
In equation (1.1), if a = 2, b = 3, k = 1 and q = 2 the Newell-Whitehead-Segel equation is written as: subject to a constant initial condition u(x, 0) = 1. Now by using variational iteration method, we have 2) whereũ n is restricted variation so δũ n = 0. Then applying variation on both sides and integrating we obtain equations Now from these equations, we get Then by VIM we can construct a sequence. Initially with the given initial condition we can take u(x, 0) = u 0 (x, t) = 1 and by VIM λ = −e −2(t−ξ) and then And also we know that Taylor's series expansion of So as n tends ∞, u n (x, t) tends to   which is the exact solution. Fig. (1 − 3) shows the comparison between exact solution and fourth order approximate solution for different intervals of time. It can be observed from Fig.  (1 − 3) that this method is effective in a narrow band of time.

Case II.
In equation (1.1), if a = 1, b = 1, k = 1 and q = 2, then the Newell-Whitehead-Segel equation is written as: subject to the initial condition  Then by using variational iteration method, we have whereũ n is restricted variation so δũ n = 0. Then applying variation on both sides and integrating we obtain equations Now from these equations, we get λ = −e −(t−ξ) Then by VIM we can construct a sequence. Initially with the given initial condition we can take which is the exact solution.

Case III.
In equation (1.1), if a = 1, b = 1, k = 1 and q = 4, the Newell-Whitehead-Segel equation is written as:   Now using variational iteration method, we have whereũ n is restricted variation so δũ n = 0. Then applying variation on both sides and integrating we obtain equations Now from these equations, we get λ = −e −(t−ξ) . Then by VIM we can construct a sequence. Initially with the given initial condition we can take So as n tends ∞, u n (x, t) tends to which is the exact solution.

Conclusion
In the present work, He's variational iteration method (VIM) has been successfully applied to obtain numerical solution for various types of Newell-Whitehead-Segel nonlinear diffusion equation. It is apparently seen that VIM is a very efficient and powerful numerical method to obtain the approximate solution. It is shown that He's variational iteration method is a promising tool for treating nonlinear equations and in some cases yields exact solution in a few iteration. The method is used in a direct way without using adomain polynomial, linearization, perturbation or restrictive assumptions. Therefore, FVIM is easier and more convenient than other methods.