Numerical Solution for Solving Burger's-fisher Eguation by Using Iterative Methods

In this paper, a Burger's-Fisher equation is solved by using the Adomian's decomposition method (ADM) , modified Adomian's decomposition method (MADM), variational iteration method (VIM), modified variational iteration method (MVIM), modified homotopy perturbation method (MHPM) and homotopy analysis method (HAM). The approximate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the presented methods. 1.INTRODUCTION Burger's-Fisher equation playes an important role in mathematical physics. In recent years some works have been done in order to find the numerical solution of this equation. For example [1-11,35-37]. In this work, we develop the ADM, MADM, VIM, MVIM, MHPM and HAM to solve the Burger's –Fisher equation as follows:   2 σ σ β 1 ,0 ,0 , 2 u u u au u u x L t T t x x               (1) with the initial condition given by:    t t a a a a t u        (3) a a u L t t t a

with the initial condition given by: and boundary conditions : Where α, β and σ are constants. When α =0,σ = 1, Eq.(1) is reduced to the Huxley equation which describes nerve pulse propagation in nerve fibre and wall motion in liquid crystals [12]. Generalized Burger equation will be obtained when β =0. This equation when β = 0, has been used to investigate sound waves in a viscous medium by Lighthill [13]. However, it was originally introduced by Burgers [14] to model one-dimensional turbulence and can also be applied to waves in fluid-filled viscous elastic tubes and magnetohydrodynamic waves in a medium with finite electrical conductivity [15].
In order to obtain an approximate solution of Eq.(1), let us integrate one time Eq.(1) with respect to t using the initial conditions we obtain, where, , , The terms D 2 (u(x,t)), F(u(x,t)) and F 1 (u(x,t)) are Lipschitz continuous with

. THE ITERATIVE METHODS 2.1 Description of the MADM and ADM
The Adomian decomposition method is applied to the following general nonlinear equation , Lu Ru Nu g    ) 6 ( where u is the unknown function, L is the highest order derivative operator which is assumed to be easily invertible, R is a linear differential operator of order less than L, Nu represents the nonlinear terms, and g is the source term. Applying the inverse operator L -1 to both sides of Eq.(6), and using the given conditions we obtain where the function f(x) represents the terms arising from integrating the source term g . The nonlinear operator Nu = G 1 (u) is decomposed as where n A , n 0  , are the Adomian polynomials determined formally as follows : Adomian polynomials were introduced in as  ,

Adomian decomposition method
The standard decomposition technique represents the solution of u(x, t) in (6) as the following series, (11) where, the components 0 u , 1 u ,... are usually determined recursively by Substituting (10) into (12) leads to the determination of the components of u. Having determined the components u 0 ,u 1 ,... the solution u in a series form defined by (11) follows immediately.

The modified Adomian decomposition method
The modified decomposition method was introduced by Wazwaz [19]. The modified forms was established based on the assumption that the function f(x) can be divided into two parts, namely f 1 (x) and f 2 (x). Under this assumption we set ) 13 (      .
2 1 x f x f x f   Accordingly, a slight variation was proposed only on the components u 0 and u 1 . The suggestion was that only the part f 1 be assigned to the zeroth component u 0 , whereas the remaining part f 2 be combined with the other terms given in (12) to define u 1 . Consequently, the modified recursive relation ) 14 ( To obtain the approximation solution of Eq.(1), according to the MADM, we can write the iterative formula ) 14 ( as follows: The operators 2 D (u), F(u) , and 1 F (u) are usually represented by the infinite series of the Adomian polynomials as follows :  are the Adomian polynomials . Also, we can use the following formula for the Adomian polynomials 20 [ ]:

Description of the VIM and MVIM
To obtain the approximation solution of Eq.(1), according to the VIM[ 24 -21 ,33-34], we can write iteration formula as follows : To find the optimal λ, we proceed as (18), the stationary conditions can be obtained as follows: Therefore, the Lagrange multipliers can be identified as λ = -1 and by substituting in (17), the following iteration formula is obtained .  (19) and (20) will enable us to determine the components u n (x, t) recursively for n ≥ 0.

Description of the HAM
where N is a nonlinear operator, u(x, t) is unknown function and x is an independent variable. Let u 0 (x,t) denote an initial guess of the exact solution u(x,t), h 0 ≠ an auxiliary parameter, H(x, t) ≠ 0 an auxiliary function, and L an auxiliary nonlinear operator with the property L[s(x,t)]= 0 when s(x, t) = 0. Then using q [0, 1] as an embedding parameter, we construct a homotopy as follows : It should be emphasized that we have great freedom to choose the initial guess u 0 (x,t) , the auxiliary nonlinear operator L, the non-zero auxiliary parameter h, and the auxiliary function H(x, t). Enforcing the homotopy (21) to be zero, i.e . , ) 22 (  , t x u t x   Thus, according to (24) and (25), as the embedding parameter q increases from 0 to 1, φ(x, t; q) varies continuously from the initial approximation u 0 (x,t) to the exact solution u(x, t). Such a kind of continuous variation is called deformation in homotopy 28, 29 .
[ ] Due to Taylor's theorem, φ(x, t;q) can be expanded in a power series of q as follows ) 26 ( Let the initial guess u 0 (x, t), the auxiliary nonlinear parameter L, the nonzero auxiliary parameter h and the auxiliary function H(x, t) be properly chosen so that the power series (26) By differentiating (28) m times with respect to q, we obtain To obtain the approximation solution of Eq.(1), according to HAM, let We take an initial guess u 0 (x,t) = f(x) , an auxiliary nonlinear operator Lu = u, a nonzero auxiliary parameter h = -1, and auxiliary function H(x,t) = 1. This is substituted into (32) to give the recurrence relation ) 33 (

F(u(x, t)) g (x)h (t), F (u(x, t)) g (x)h (t) and D (u(x, t)) g (x)h (t).
   We can define homotopy H(u(x, t), p, m) by Where m is an unknown real number and where m is called the accelerating parameters, and for m=0 we define H(u(x, t,  p,0) = H(u(x, t),p), which is the standard HPM .
The convex homotopy (34)   Proof. Let u and u * be two different solutions of (5) then
and completes the proof . Define the sequence of partial sums s n , let s n and s m be arbitrary partial sums with n ≥ m. We are going to prove that s n is a Cauchy sequence in this Banach space: We conclude that s n is a Cauchy sequence in C[J], therefore the series is convergence and the proof is complete . Theorem 3.3. The series solution ( , ) u x t n of problem (1) using VIM converges when

(
By subtracting relation (38) from (39), x t u x t u x t e x t u x t u x t e x t e x t n n n n n n t n then since e n is a decreasing function with respect to t from the mean value theorem we can write , J e e n n n t       Since 0 0, 0 1 then e n     . So, the series converges and the proof is complete .
Theorem 3.4. The series solution ( , ) u x t n of problem (1) using MVIM converges when 0< α 1 < 1 , 0< β 2 < 1 . Proof. The proof is similar to the previous theorem. Theorem 3.5. If the series solution (33) of problem (1) using HAM convergent then it converges to the exact solution of the problem (1). Proof.. We assume :  x u x   With the exact solution is 1

CONCLUSION
The MHPM has been shown to solve effectively, easily and accurately a large class of nonlinear problems with the approximations which convergent are rapidly to exact solutions. In this work, the MHPM has been successfully employed to obtain the approximate analytical solution of the Burger's-Fisher equation. For this purpose, we showed that the MHPM is more rapid convergence than the ADM, MADM, VIM, MVIM and HAM .