Numerical Solution of Allen – Cahn Equation by Adomian Decomposition Method

In this paper, Adomian Decomposition Method with Adomian polynomials are applied to solve Allen Cahn equation with the initial condition only, also DuFortFrankel method is applied with the initial and boundary conditions. The numerical results that are obtained by the Adomian decomposition method have been compared with the exact solution of the equation shown that it is more efficient than the DuFortFrankel method, that is illustrated through the tables and Figures.


Introduction
Allen-Cahn equation is origin in the gradient theory of phase transitions [8], this equation is a model in which two distinct phases (represented by the values 1  = u ) try to coexist in a domain  while minimizing their interaction which is proportional to the (N-1)-dimension volume of the inter face.
Jeong-Whan Choi, et. al. in 2001, they used an unconditionally gradient table scheme for solving the Allen-Cahn equation representing a model for anti-phase domain coarsening in a binary mixture [19].
In 2011, Michal , et. al., introduced a numerical scheme allowing to perform computational studies of the anisotropic nonlocal Allen-Cahn equation [22].
The numerical solution of the generalized Allen-Cahn equation is used by Michal et. al. in 2004,[23] that proposed an algorithm for image segmentation, this technique is devoted to recovery of pattern boundaries from the original, possibly noisy image or signal.
Allen-Cahn equation, is a special type of non-linear partial differential equation, which arises as diffusionconvection equation in computational fluid dynamics or reactiondiffusion problem in material science. These equations are originally used to solve the phase transition problems, transformation of thermodynamic system from one phase to another due to an abrupt change in one or more physical properties. These equations describe the evolution of a diffuse phase boundary, concentrated in a small origin of size  . It is also arising from phase transition in materials science [5,6]. Ali, et. al. in 2011, used some numerical methods for solving Allen-Cahn equation by different time stepping and space discretization methods with non-periodic boundary condition such as Chebyshov spectral method, fourth-order Runge-Kutta method, implicit-explicit scheme and finite difference scheme [6].
In the end of the years of the last century, Adomian [2,3,4] added a new method that solves differential equations, which is an efficient for analytic and numerical solution. Ismail, et. al. in 2004, [16] studied the approximate solution for the Burger's -Huxley and Burger's -Fisher equations and in [11] studied the numerical solutions of the Korteweg -De -Vries (KDV) and modified Korteweg -De -Vries (MKDV) equations, applied Adomian decomposition method.
Adomian decomposition method is also used to approximate the solution for the differential-algebraic equations (DAEs) systems [10,15].
Alabdulatifet. al. in 2007, [5] used Adomian decomposition method to find an analytic approximate solution for non-linear reaction diffusion system of Lotk -Volterra type.
Javidi and Golbabai [18] used Adomian decomposition method for solving the linear and nonlinear parabolic equations is implemented with appropriate initial conditions and they show that is an efficient method. Bokhari, et.al.in 2009, [9] shows that the Adomian series solution gives an excellent approximation to the exact solution for non-linear heat equation with temperature dependent thermal properties.
Ghoreishi, et. al. in 2010, [14] used ADM to solve non-linear wavelike equations with variable coefficients and they show that ADM is able to solve this type of equations without any need for dissertation, perturbation, transformation and linearization.
Abdelwahid, presented Adomian decomposition method is accuracy, applicability and simplicity to solve non-linear integral equation [1].
Space -time fractional telegraph equation is solved by Adomian decomposition method [13]. Cheniguel and Ayadi [12] presented the Adomian decomposition method for solving non homogenous heat equation with an initial condition and non local boundary conditions and they show that this technique is an efficient of the other classical methods.
Montazeri, applied Adomian decomposition method to solve a system of partial differential equations and show that the solutions, results are efficient [24].
Al-Khaled and Allan [7] are gave outlines a reliable strategy for solving nonlinear Volterra -Fredholmintegrodifferential equations by (ADM) and from numerical examples they presented to illustratethe accuracy of the method.
Mamloukas, find an approximate solution of Burger's equation using the decomposition method [21].
In this research, we apply Adomian decomposition method and DuFort-Frankel method to Allen -Cahn equation [20]:

2.Analysis of TheAdomian Decomposition Method (ADM)
In this section, we review the main steps of Adomian decomposition method on linear and non-linear parabolic partial differential equations with initial condition only, consider [30]: Where, F is a non-linear function of u. We find the solution satisfying eqs. (4)-(5). Adomian decomposition method assumes that the unknown function u(x,t) can be expressed by an infinite series of the form: And the non-linear operator F(u) can be decomposed by an infinite series of polynomial given by It is now well known in the literature, that these polynomials can be constructed for all classes of non-linearity according to algorithm set by Adomian [3,4] and recently developed by an alternative approach in [2,16,17,26,27,28,29] .
Then, the solution series of the equation that we get by ADM leads to the exact solution of u(x,t) when k is an approach to infinity, i.e.

Application of ADM to Allen-Cahn Equation
In this section, we apply ADM to the Allen-Cahn eq. (1) with an initial condition in eq.(2).
Appling the inverse operator 1 − L on both sides of eq.(1) and using the initial condition, yields   Where, An are Adomian polynomial that represent the non-linear term (u 3 ), then we can obtain from eq.(8) as follows: Then, the solutions are obtained by using the initial condition only. Thus, we begin:

Application of DFM to Allen-Cahn Equation
In this section, we apply DFM for Allen-Cahn eq. (1) with the initial condition in eq.(2) and boundary conditions in eq.(3). DFM is approximated the partial derivatives with respect to time t and space x with the second order accuracy through centered differencing formula, then eq.(1), by this method, leads to [25]:
Then, the numerical results of (AC) equation by exact solution, ADM, and DFM are shown in table (1) (1) and (2) shown the surface of the numerical solution u(x,t) of (AC) equation when In order to show that numerically whether the Adomian decomposition for eq.(1) leads to a higher accuracy, we evaluate the approximate solution by using more than 3 terms approximations.

Conclusion
We note that the Adomian decomposition method is an efficient method when used Adomian polynomials to the treatment of the non-linear term in Allen -Cahn equation with convergence to the exact solution and small error. We see that the ADM has been applied directly without using bilinear forms. Also, we see that when we take added terms from the series solution, the ADM convergence is more efficient to the exact solution than the DFM. We used Matlab codes to obtain the solutions.