# Convergence of Iterative Methods Applied to Generalized Fisher Equation

Authors: Sh. Sadigh Behzadi
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

A generalized Fisher's equation is solved by using the modified Adomian decomposition
method (MADM), variational iteration method (VIM), homotopy analysis method (HAM), and modified homotopy
perturbation method (MHPM). The approximation solution of this equation is calculated in the form
of series whose components are computed easily. The existence, uniqueness, and convergence of the proposed
methods are proved. Numerical example is studied to demonstrate the accuracy of the present methods.


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## Body

## 1. Introduction

Fisher proposed equation∂u/∂t=∂2u/∂x2+u(1-u) as a model for the propagation of a mutant gene, with u denoting the density of an advantageous. This equation is encountered in chemical kinetics [1] and population dynamics which includes problems such as nonlinear evolution of a population in a nuclear reaction and branching. Moreover, the same equation occurs in logistic population growth models [2], flame propagation, neurophysiology, autocatalytic chemical reaction, and branching Brownian motion processes. A lot of works have been done in order to find the numerical solution of this equation, for example, variational iteration method and modified variational iteration method for solving the generalized Fisher equation [3–5], an analytical study of Fisher equation by using Adomian decomposition method [6], numerical solution for solving Burger-Fisher equation [7–10], a novel approach for solving the Fisher equation using Exp-function method [11]. In this paper, we develop the MADM, VIM, HAM, and MHPM to solve the generalized Fisher equation as follows:
(1.1)∂u∂t=∂2u∂x2+u(1-us),
with the initial conditions given by(1.2)u(x,0)=f(x).The paper is organized as follows. In Section2, the iteration methods MADM, VIM, HAM and MHPM are introduced for solving (1.1). Also, the existance, uniqueness and convergence of the proposed in Section 3. Finally, the numerical example is presented in Section 4 to illustrate the accuracy of these methods.To obtain the approximation solution of (1.1), by integrating one time from (1.1) with respect to t and using the initial conditions, we obtain(1.3)u(x,t)=f(x)+∫0t∂2u(x,τ)∂x2dτ+∫0tu(x,τ)(1-us(x,τ))dτ.We set(1.4)F(u)=u(1-us).In (1.3), we assume f(x) is bounded for all x in J=[0,T](T∈ℝ) and(1.5)|t-τ|≤M′,∀0≤t,τ≤T.The termsD2(u) and F(u) are Lipschitz continuous with |D2(u)-D2(u*)|≤L1|u-u*|, |F(u)-F(u*)|≤L2|u-u*|.We set(1.6)α=T(M′L1+M′L2),β=1-T(1-α).Now we decompose the unknown functionu(x,t) by a sum of components defined by the following decomposition series with u0 identified as u(x,0):(1.7)u(x,t)=∑n=0∞un(x,t).


## 2. Iterative Methods

### 2.1. Preliminaries of the MADM

The Adomian decomposition method is applied to the following general nonlinear equation:(2.1)Lu+Ru+Nu=g(x),
where u is the unknown function, L is the highest-order derivative 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 (2.1), and using the given conditions, we obtain
(2.2)u=f(x)-L-1(Ru)-L-1(Nu),
where the function f(x) represents the terms arising from integrating the source term g(x). The nonlinear operator Nu=G(u) is decomposed as(2.3)G(u)=∑n=0∞An,
where An, n≥0 are the Adomian polynomials determined formally as follows:(2.4)An=1n![dndλn[N(∑i=0∞λiui)]]λ=0.
Adomian polynomials were introduced in [12–15] as(2.5)A0=G(u0),A1=u1G′(u0),A2=u2G′(u0)+12!u12G′′(u0),A3=u3G′(u0)+u1u2G′′(u0)+13!u13G′′′(u0),….


#### 2.1.1. Adomian Decomposition Method

The standard decomposition technique represents the solution ofu in (2.1) as the following series,(2.6)u=∑n=0∞ui,
where, the components u0,u1,… are usually determined recursively by(2.7)u0=f(x),un+1=-L-1(Run)-L-1(An),n≥0.
Substituting (2.5) into (2.7) leads to the determination of the components of u. Having determined the components u0,u1,… the solution u in a series form defined by (2.6) follows immediately.


#### 2.1.2. The Modified Adomian Decomposition Method

The modified decomposition method was introduced by Wazwaz in [16]. The modified forms was established based on the assumption that the function f(x) can be divided into two parts, namely f1(x) and f2(x). Under this assumption we set(2.8)f(x)=f1(x)+f2(x).
Accordingly, a slight variation was proposed only on the components u0 and u1. The suggestion was that only the part f1 be assigned to the zeroth component u0, whereas the remaining part f2 be combined with the other terms given in (2.7) to define u1. Consequently, the modified recursive relation(2.9)u0=f1(x),u1=f2(x)-L-1(Ru0)-L-1(A0),⋮un+1=-L-1(Run)-L-1(An),n≥1,
was developed.


### 2.2. Description of the MADM

To obtain the approximation solution of (1.1), according to the MADM, we can write the iterative formula (2.9) as follows:(2.10)u0(x,t)=f1(x),u1(x,t)=f2(x)+∫0tD2(u0(x,τ))+∫0tF(u0(x,τ))dτ,⋮un+1(x,t)=∫0tD2(un(x,τ))+∫0tF(un(x,τ))dτ.The operatorsD2(u(x,τ))=(d2/dx2)u(x,t) and F(u(x,τ)) are usually represented by an infinite series of the so-called Adomian polynomials as follows:(2.11)F(u)=∑i=0∞Ai,D2(u)=∑i=0∞Li.
where Ai and Li(i≥0) are the Adomian polynomials were introduced in [12].From [12], we can write another formula for the Adomian polynomials:(2.12)Ln=D2(sn)-∑i=0n-1Li,An=F(sn)-∑i=0n-1Ai,
where the partial sum is sn=∑i=0nui(x,t).


### 2.3. Preliminaries of the VIM

In the VIM [17–20], we consider the following nonlinear differential equation:(2.13)L(u)+N(u)=g(t),
where L is a linear operator,N is a nonlinear operator and g(x,t) is a known analytical function. In this case, a correction functional can be constructed as follows:(2.14)un+1(x,t)=un(x,t)+∫0tλ(x,τ){L(un(x,τ))+N(un(x,τ))-g(x,τ)}dτ,n≥0,
where λ is a general Lagrange multiplier which can be identified optimally via variational theory. Here the function un(x,τ) is a restricted variations which means δun=0. Therefore, we first determine the Lagrange multiplier λ that will be identified optimally via integration by parts. The successive approximation un(x,t), n≥0 of the solution u(t) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function u0. The zeroth approximation u0 may be selected any function that just satisfies at least the initial and boundary conditions. With λ determined, then several approximation un(x,t), n≥0 follow immediately. Consequently, the exact solution may be obtained by using
(2.15)u(x,t)=lim⁡n→∞un(x,t).The VIM has been shown to solve effectively, easily and accurately a large class of nonlinear problems with approximations converge rapidly to accurate solutions.


### 2.4. Description of the VIM

To obtain the approximation solution of (1.1), according to the VIM, we can write iteration formula (2.14) as follows:
(2.16)un+1(x,t)=un(x,t)+Lt-1(λ[u(x,t)-f(x)-∫0tD2(un(x,τ))dτ-∫0tF(un(x,τ))dτ]),
where,
(2.17)Lt-1(⋅)=∫0t(⋅)dτ.To find the optimalλ, we proceed as(2.18)δun+1(x,t)=δun(x,t)+δLt-1(λ[un(x,t)-f(x)-∫0tD2(un(x,τ))dτ+∫0tF(un(x,τ))dτ])=δun(x,t)+λ(x)δun(x,t)-Lt-1[δun(x,t)λ′(x)].From (2.18), the stationary conditions can be obtained as follows:(2.19)λ′=0,1+λ=0.Therefore, the Lagrange multipliers can be identified asλ=-1 and by substituting in (2.16), the following iteration formula is obtained.(2.20)u0(x,t)=f(x),un+1(x,t)=un(x,t)-Lt-1([un(x,t)-f(x)-∫0tD2(un(x,τ))dτ-∫0tF(un(x,τ))dτ]),n≥0.Relation (2.20) will enable us to determine the components un(x,t) recursively for n≥0.


### 2.5. Preliminaries of the HAM

Consider(2.21)N[u]=0,
where N is a nonlinear operator, u(x,t) is unknown function and x is an independent variable. let u0(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[r(x,t)]=0 when r(x,t)=0. Then using q∈[0,1] as an embedding parameter, we construct a homotopy as follows:(2.22)(1-q)L[ϕ(x,t;q)-u0(x,t)]-qhH(x,t)N[ϕ(x,t;q)]=Ĥ[ϕ(x,t;q);u0(x,t),H(x,t),h,q].It should be emphasized that we have great freedom to choose the initial guessu0(x,t), the auxiliary nonlinear operator L, the nonzero auxiliary parameter h, and the auxiliary function H(x,t).Enforcing the homotopy (2.22) to be zero, that is,
(2.23)Ĥ[ϕ(x,t;q);u0(x,t),H(x,t),h,q]=0,we have the so-called zero-order deformation equation(2.24)(1-q)L[ϕ(x,t;q)-u0(x,t)]=qhH(x,t)N[ϕ(x,t;q)].Whenq=0, the zero-order deformation (2.24) becomes(2.25)ϕ(x,t;0)=u0(x,t),
and when q=1, since h≠0 and H(x,t)≠0, the zero-order deformation (2.24) is equivalent to(2.26)ϕ(x,t;1)=u(x,t).Thus, according to (2.25) and (2.26), as the embedding parameter q increases from 0 to 1, ϕ(x,t;q) varies continuously from the initial approximation u0(x,t) to the exact solution u(x,t). Such a kind of continuous variation is called deformation in homotopy [21, 22].Due to Taylor's theorem,ϕ(x,t;q) can be expanded in a power series of q as follows:(2.27)ϕ(x,t;q)=u0(x,t)+∑m=1∞um(x,t)qm,
where(2.28)um(x,t)=1m!∂mϕ(x,t;q)∂qm|q=0.Let the initial guessu0(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 (2.27) of ϕ(x,t;q) converges at q=1, then, we have under these assumptions the solution series(2.29)u(x,t)=ϕ(x,t;1)=u0(x,t)+∑m=1∞um(x,t).From (2.27), we can write (2.24) as follows:(2.30)(1-q)L[ϕ(x,t;q)-u0(x,t)]=(1-q)L[∑m=1∞um(x,t)qm]=qhH(x,t)N[ϕ(x,t;q)]⟹L[∑m=1∞um(x,t)qm]-qL[∑m=1∞um(x,t)qm]=qhH(x,t)N[ϕ(x,t;q)].By differentiating (2.30) m times with respect to q, we obtain(2.31){L[∑m=1∞um(x,t)qm]-qL[∑m=1∞um(x,t)qm]}(m)={qhH(x,t)N[ϕ(x,t;q)]}(m)=m!L[um(x,t)-um-1(x,t)]=hH(x,t)m∂m-1N[ϕ(x,t;q)]∂qm-1|q=0.Therefore,(2.32)L[um(x,t)-χmum-1(x,t)]=hH(x,t)Rm(ym-1(x)),
where,(2.33)Rm(um-1(x,t))=1(m-1)!∂m-1N[ϕ(x,t;q)]∂qm-1|q=0,(2.34)χm={0,m≤1,1,m>1.Note that the high-order deformation (2.32) is governing the nonlinear operator L, and the term ℜm(um-1(x,t)) can be expressed simply by (2.33) for any nonlinear operator N.


### 2.6. Description of the HAM

To obtain the approximation solution of (1.1), according to HAM, let(2.35)N[u]=u(x,t)-f(x)-∫0tD2(u(x,τ))dτ-∫0tF(u(x,τ))dτ,
so
(2.36)Rm(um-1(x,t))=um-1(x,t)-∫0tD2(um-1(x,τ))dτ-∫0tF(um-1(x,τ))dτ-(1-χm)f(x).Substituting (2.36) into(2.32)(2.37)L[um(x,t)-χmum-1(x,t)]=hH(x,t)[um-1(x,t)-∫0tD2(um-1(x,τ))dτ-∫0tF(um-1(x,τ))dτ-(1-χm)f(x)].We take an initial guessu0(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 (2.37) to give the recurrence relation
(2.38)u0(x,t)=f(x),un(x,t)=∫0tD2(un-1(x,τ))dτ+∫0tF(un-1(x,τ))dτ,n≥1.Therefore, the solutionu(x,t) becomes(2.39)u(x,t)=∑n=0∞un(x,t)=f(x)+∑n=0∞(∫0tD2(un-1(x,τ))dτ+∫0tF(un-1(x,τ))dτ),
which is the method of successive approximations. If(2.40)|un(x,t)|<1
then the series solution (2.39) convergence uniformly.


### 2.7. Description of the MHPM

To explain MHPM, we consider (1.1) as(2.41)L(u)=u(x,t)-f(x)-∫0tD2(un-1(x,τ))dτ-∫0tF(un-1(x,τ))dτ,
where D2(u(x,τ))=g1(x)h1(τ) and F(u(x,τ))=g2(x)h2(τ). We can define homotopy H(u,p,m) by(2.42)H(u,o,m)=f(u),H(u,1,m)=L(u),
where m is an unknown real number and(2.43)f(u(x,t))=u(x,t)-G(x,t).
Typically we may choose a convex homotopy by(2.44)H(u,p,m)=(1-p)f(u)+pL(u)+p(1-p)[m(g1(x)+g2(x))]=0,0≤p≤1.
where m is called the accelerating parameters, and for m=0 we define H(u,p,0)=H(u,p), which is the standard HPM. The convex homotopy (2.44) continuously trace an implicity defined curve from a starting point H(u(x,t)-f(u),0,m) to a solution function H(u(x,t),1,m). The embedding parameter p monotonically increase from o to 1 as trivial problem f(u)=0 is continuously deformed to original problem L(u)=0. [23, 24]The MHPM uses the homotopy parameterp as an expanding parameter to obtain(2.45)v=∑n=0∞pnun;
when p→1 (2.44) corresponds to the original one, (2.45) becomes the approximate solution of (1.1), that is,(2.46)u=lim⁡p→1v=∑m=0∞um,
where,(2.47)um(x,t)=f(x)+∫0tD2(um-1(x,τ))dτ+∫0tF(um-1(x,τ))dτ.


## 2.1. Preliminaries of the MADM

The Adomian decomposition method is applied to the following general nonlinear equation:(2.1)Lu+Ru+Nu=g(x),
where u is the unknown function, L is the highest-order derivative 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 (2.1), and using the given conditions, we obtain
(2.2)u=f(x)-L-1(Ru)-L-1(Nu),
where the function f(x) represents the terms arising from integrating the source term g(x). The nonlinear operator Nu=G(u) is decomposed as(2.3)G(u)=∑n=0∞An,
where An, n≥0 are the Adomian polynomials determined formally as follows:(2.4)An=1n![dndλn[N(∑i=0∞λiui)]]λ=0.
Adomian polynomials were introduced in [12–15] as(2.5)A0=G(u0),A1=u1G′(u0),A2=u2G′(u0)+12!u12G′′(u0),A3=u3G′(u0)+u1u2G′′(u0)+13!u13G′′′(u0),….


### 2.1.1. Adomian Decomposition Method

The standard decomposition technique represents the solution ofu in (2.1) as the following series,(2.6)u=∑n=0∞ui,
where, the components u0,u1,… are usually determined recursively by(2.7)u0=f(x),un+1=-L-1(Run)-L-1(An),n≥0.
Substituting (2.5) into (2.7) leads to the determination of the components of u. Having determined the components u0,u1,… the solution u in a series form defined by (2.6) follows immediately.


### 2.1.2. The Modified Adomian Decomposition Method

The modified decomposition method was introduced by Wazwaz in [16]. The modified forms was established based on the assumption that the function f(x) can be divided into two parts, namely f1(x) and f2(x). Under this assumption we set(2.8)f(x)=f1(x)+f2(x).
Accordingly, a slight variation was proposed only on the components u0 and u1. The suggestion was that only the part f1 be assigned to the zeroth component u0, whereas the remaining part f2 be combined with the other terms given in (2.7) to define u1. Consequently, the modified recursive relation(2.9)u0=f1(x),u1=f2(x)-L-1(Ru0)-L-1(A0),⋮un+1=-L-1(Run)-L-1(An),n≥1,
was developed.


## 2.1.1. Adomian Decomposition Method

The standard decomposition technique represents the solution ofu in (2.1) as the following series,(2.6)u=∑n=0∞ui,
where, the components u0,u1,… are usually determined recursively by(2.7)u0=f(x),un+1=-L-1(Run)-L-1(An),n≥0.
Substituting (2.5) into (2.7) leads to the determination of the components of u. Having determined the components u0,u1,… the solution u in a series form defined by (2.6) follows immediately.


## 2.1.2. The Modified Adomian Decomposition Method

The modified decomposition method was introduced by Wazwaz in [16]. The modified forms was established based on the assumption that the function f(x) can be divided into two parts, namely f1(x) and f2(x). Under this assumption we set(2.8)f(x)=f1(x)+f2(x).
Accordingly, a slight variation was proposed only on the components u0 and u1. The suggestion was that only the part f1 be assigned to the zeroth component u0, whereas the remaining part f2 be combined with the other terms given in (2.7) to define u1. Consequently, the modified recursive relation(2.9)u0=f1(x),u1=f2(x)-L-1(Ru0)-L-1(A0),⋮un+1=-L-1(Run)-L-1(An),n≥1,
was developed.


## 2.2. Description of the MADM

To obtain the approximation solution of (1.1), according to the MADM, we can write the iterative formula (2.9) as follows:(2.10)u0(x,t)=f1(x),u1(x,t)=f2(x)+∫0tD2(u0(x,τ))+∫0tF(u0(x,τ))dτ,⋮un+1(x,t)=∫0tD2(un(x,τ))+∫0tF(un(x,τ))dτ.The operatorsD2(u(x,τ))=(d2/dx2)u(x,t) and F(u(x,τ)) are usually represented by an infinite series of the so-called Adomian polynomials as follows:(2.11)F(u)=∑i=0∞Ai,D2(u)=∑i=0∞Li.
where Ai and Li(i≥0) are the Adomian polynomials were introduced in [12].From [12], we can write another formula for the Adomian polynomials:(2.12)Ln=D2(sn)-∑i=0n-1Li,An=F(sn)-∑i=0n-1Ai,
where the partial sum is sn=∑i=0nui(x,t).


## 2.3. Preliminaries of the VIM

In the VIM [17–20], we consider the following nonlinear differential equation:(2.13)L(u)+N(u)=g(t),
where L is a linear operator,N is a nonlinear operator and g(x,t) is a known analytical function. In this case, a correction functional can be constructed as follows:(2.14)un+1(x,t)=un(x,t)+∫0tλ(x,τ){L(un(x,τ))+N(un(x,τ))-g(x,τ)}dτ,n≥0,
where λ is a general Lagrange multiplier which can be identified optimally via variational theory. Here the function un(x,τ) is a restricted variations which means δun=0. Therefore, we first determine the Lagrange multiplier λ that will be identified optimally via integration by parts. The successive approximation un(x,t), n≥0 of the solution u(t) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function u0. The zeroth approximation u0 may be selected any function that just satisfies at least the initial and boundary conditions. With λ determined, then several approximation un(x,t), n≥0 follow immediately. Consequently, the exact solution may be obtained by using
(2.15)u(x,t)=lim⁡n→∞un(x,t).The VIM has been shown to solve effectively, easily and accurately a large class of nonlinear problems with approximations converge rapidly to accurate solutions.


## 2.4. Description of the VIM

To obtain the approximation solution of (1.1), according to the VIM, we can write iteration formula (2.14) as follows:
(2.16)un+1(x,t)=un(x,t)+Lt-1(λ[u(x,t)-f(x)-∫0tD2(un(x,τ))dτ-∫0tF(un(x,τ))dτ]),
where,
(2.17)Lt-1(⋅)=∫0t(⋅)dτ.To find the optimalλ, we proceed as(2.18)δun+1(x,t)=δun(x,t)+δLt-1(λ[un(x,t)-f(x)-∫0tD2(un(x,τ))dτ+∫0tF(un(x,τ))dτ])=δun(x,t)+λ(x)δun(x,t)-Lt-1[δun(x,t)λ′(x)].From (2.18), the stationary conditions can be obtained as follows:(2.19)λ′=0,1+λ=0.Therefore, the Lagrange multipliers can be identified asλ=-1 and by substituting in (2.16), the following iteration formula is obtained.(2.20)u0(x,t)=f(x),un+1(x,t)=un(x,t)-Lt-1([un(x,t)-f(x)-∫0tD2(un(x,τ))dτ-∫0tF(un(x,τ))dτ]),n≥0.Relation (2.20) will enable us to determine the components un(x,t) recursively for n≥0.


## 2.5. Preliminaries of the HAM

Consider(2.21)N[u]=0,
where N is a nonlinear operator, u(x,t) is unknown function and x is an independent variable. let u0(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[r(x,t)]=0 when r(x,t)=0. Then using q∈[0,1] as an embedding parameter, we construct a homotopy as follows:(2.22)(1-q)L[ϕ(x,t;q)-u0(x,t)]-qhH(x,t)N[ϕ(x,t;q)]=Ĥ[ϕ(x,t;q);u0(x,t),H(x,t),h,q].It should be emphasized that we have great freedom to choose the initial guessu0(x,t), the auxiliary nonlinear operator L, the nonzero auxiliary parameter h, and the auxiliary function H(x,t).Enforcing the homotopy (2.22) to be zero, that is,
(2.23)Ĥ[ϕ(x,t;q);u0(x,t),H(x,t),h,q]=0,we have the so-called zero-order deformation equation(2.24)(1-q)L[ϕ(x,t;q)-u0(x,t)]=qhH(x,t)N[ϕ(x,t;q)].Whenq=0, the zero-order deformation (2.24) becomes(2.25)ϕ(x,t;0)=u0(x,t),
and when q=1, since h≠0 and H(x,t)≠0, the zero-order deformation (2.24) is equivalent to(2.26)ϕ(x,t;1)=u(x,t).Thus, according to (2.25) and (2.26), as the embedding parameter q increases from 0 to 1, ϕ(x,t;q) varies continuously from the initial approximation u0(x,t) to the exact solution u(x,t). Such a kind of continuous variation is called deformation in homotopy [21, 22].Due to Taylor's theorem,ϕ(x,t;q) can be expanded in a power series of q as follows:(2.27)ϕ(x,t;q)=u0(x,t)+∑m=1∞um(x,t)qm,
where(2.28)um(x,t)=1m!∂mϕ(x,t;q)∂qm|q=0.Let the initial guessu0(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 (2.27) of ϕ(x,t;q) converges at q=1, then, we have under these assumptions the solution series(2.29)u(x,t)=ϕ(x,t;1)=u0(x,t)+∑m=1∞um(x,t).From (2.27), we can write (2.24) as follows:(2.30)(1-q)L[ϕ(x,t;q)-u0(x,t)]=(1-q)L[∑m=1∞um(x,t)qm]=qhH(x,t)N[ϕ(x,t;q)]⟹L[∑m=1∞um(x,t)qm]-qL[∑m=1∞um(x,t)qm]=qhH(x,t)N[ϕ(x,t;q)].By differentiating (2.30) m times with respect to q, we obtain(2.31){L[∑m=1∞um(x,t)qm]-qL[∑m=1∞um(x,t)qm]}(m)={qhH(x,t)N[ϕ(x,t;q)]}(m)=m!L[um(x,t)-um-1(x,t)]=hH(x,t)m∂m-1N[ϕ(x,t;q)]∂qm-1|q=0.Therefore,(2.32)L[um(x,t)-χmum-1(x,t)]=hH(x,t)Rm(ym-1(x)),
where,(2.33)Rm(um-1(x,t))=1(m-1)!∂m-1N[ϕ(x,t;q)]∂qm-1|q=0,(2.34)χm={0,m≤1,1,m>1.Note that the high-order deformation (2.32) is governing the nonlinear operator L, and the term ℜm(um-1(x,t)) can be expressed simply by (2.33) for any nonlinear operator N.


## 2.6. Description of the HAM

To obtain the approximation solution of (1.1), according to HAM, let(2.35)N[u]=u(x,t)-f(x)-∫0tD2(u(x,τ))dτ-∫0tF(u(x,τ))dτ,
so
(2.36)Rm(um-1(x,t))=um-1(x,t)-∫0tD2(um-1(x,τ))dτ-∫0tF(um-1(x,τ))dτ-(1-χm)f(x).Substituting (2.36) into(2.32)(2.37)L[um(x,t)-χmum-1(x,t)]=hH(x,t)[um-1(x,t)-∫0tD2(um-1(x,τ))dτ-∫0tF(um-1(x,τ))dτ-(1-χm)f(x)].We take an initial guessu0(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 (2.37) to give the recurrence relation
(2.38)u0(x,t)=f(x),un(x,t)=∫0tD2(un-1(x,τ))dτ+∫0tF(un-1(x,τ))dτ,n≥1.Therefore, the solutionu(x,t) becomes(2.39)u(x,t)=∑n=0∞un(x,t)=f(x)+∑n=0∞(∫0tD2(un-1(x,τ))dτ+∫0tF(un-1(x,τ))dτ),
which is the method of successive approximations. If(2.40)|un(x,t)|<1
then the series solution (2.39) convergence uniformly.


## 2.7. Description of the MHPM

To explain MHPM, we consider (1.1) as(2.41)L(u)=u(x,t)-f(x)-∫0tD2(un-1(x,τ))dτ-∫0tF(un-1(x,τ))dτ,
where D2(u(x,τ))=g1(x)h1(τ) and F(u(x,τ))=g2(x)h2(τ). We can define homotopy H(u,p,m) by(2.42)H(u,o,m)=f(u),H(u,1,m)=L(u),
where m is an unknown real number and(2.43)f(u(x,t))=u(x,t)-G(x,t).
Typically we may choose a convex homotopy by(2.44)H(u,p,m)=(1-p)f(u)+pL(u)+p(1-p)[m(g1(x)+g2(x))]=0,0≤p≤1.
where m is called the accelerating parameters, and for m=0 we define H(u,p,0)=H(u,p), which is the standard HPM. The convex homotopy (2.44) continuously trace an implicity defined curve from a starting point H(u(x,t)-f(u),0,m) to a solution function H(u(x,t),1,m). The embedding parameter p monotonically increase from o to 1 as trivial problem f(u)=0 is continuously deformed to original problem L(u)=0. [23, 24]The MHPM uses the homotopy parameterp as an expanding parameter to obtain(2.45)v=∑n=0∞pnun;
when p→1 (2.44) corresponds to the original one, (2.45) becomes the approximate solution of (1.1), that is,(2.46)u=lim⁡p→1v=∑m=0∞um,
where,(2.47)um(x,t)=f(x)+∫0tD2(um-1(x,τ))dτ+∫0tF(um-1(x,τ))dτ.


## 3. Existence and Convergency of Iterative Methods

Theorem 3.1.
Let0<α<1, then Fisher equation (1.1), has a unique solution.Proof.
Letu and u* be two different solutions of (1.3) then
(3.1)|u-u*|=|∫0t[D2(u(x,τ))-D2(u*(x,τ))]dτ+∫0t[F(u(x,τ))-F(u*(x,τ))]dτ|≤∫0t|D2(u(x,τ))-D2(u*(x,τ))|dτ+∫0t|F(u(x,τ))-F(u*(x,τ))|dτ≤T(M′L1+M′L2)|u-u*|=α|u-u*|.
From which we get(1-α)|u-u*|≤0. Since 0<α<1. then |u-u*|=0. Implies u=u* and completes the proof.Theorem 3.2.
The series solutionu(x,t)=∑i=0∞ui(x,t) of problem (1.1) using MADM convergence when 0<α<1,  |u1(x,t)|<∞.Proof.
Denote as(C[J],∥·∥) the Banach space of all continuous functions on J with the norm ∥f(t)∥=max|f(t)|, for all t in J. Define the sequence of partial sums sn, and let sn and sm be arbitrary partial sums with n≥m. We are going to prove that sn is a Cauchy sequence in this Banach space:
(3.2)‖sn-sm‖=max⁡∀t∈J⁡|sn-sm|=max⁡∀t∈J⁡|∑i=m+1nui(x,t)|=max⁡∀t∈J⁡|∑i=m+1n∫0tLi-1dτ+∑i=m+1n∫0tAi-1dτ|=max⁡∀t∈J⁡|∫0t(∑i=mn-1Li)dτ+∫0t(∑i=mn-1Ai)dτ|.
From [12], we have(3.3)∑i=mn-1Li=D2(sn-1)-D2(sm-1),∑i=mn-1Ai=F(sn-1)-F(sm-1).
So,(3.4)‖sn-sm‖=max⁡∀t∈J⁡|∫0t[D2(sn-1)-D2(sm-1)]dτ+∫0t[F(sn-1)-F(sm-1)]dτ|≤∫0t|D2(sn-1)-D2(sm-1)|dτ+∫0t|F(sn-1)-F(sm-1)|dτ≤α‖sn-sm‖.
Letn=m+1, then(3.5)‖sn-sm‖≤α‖sm-sm-1‖≤α2‖sm-1-sm-2‖≤⋯≤αm‖s1-s0‖.
From the triangle inquality we have(3.6)‖sn-sm‖≤‖sm+1-sm‖+‖sm+2-sm+1‖+⋯+‖sn-sn-1‖≤[αm+αm1+⋯+αn-m-1]‖s1-s0‖≤αm[1+α+α2+⋯+αn-m-1]‖s1-s0‖≤[1-αn-m1-α]‖u1(x,t)‖.
Since0<α<1, we have (1-αn-m)<1, then(3.7)‖sn-sm‖≤αm1-αmax⁡∀t∈J⁡|u1(x,t)|.
But|u1(x,t)|<∞, so, as m→∞, then ∥sn-sm∥→0. We conclude that sn is a Cauchy sequence in C[J], therefore the series is convergence and the proof is complete.Theorem 3.3.
The series solutionu(x,t)=∑i=0∞ui(x,t) of problem (1.1) using VIM converges when 0<α<1, 0<β<1.Proof.
One has the following:(3.8)un+1(x,t)=un(x,t)-Lt-1([un(x,t)-f(x)-∫0tD2(un(x,τ))dτ-∫0tF(un(x,τ))dτ]),(3.9)u(x,t)=u(x,t)-Lt-1([u(x,t)-f(x)-∫0tD2(u(x,τ))dτ-∫0tF(u(x,τ))dτ]).
By subtracting relation (3.9) from (3.8),(3.10)un+1(x,t)-u(x,t)=un(x,t)-u(x,t)-Lt-1(un(x,t)-u(x,t)-∫0t[D2(un(x,τ))-D2(u(x,τ))]dτ-∫0t[F(un(x,τ))-F(u(x,τ))]dτ),
if we set, en+1(r,t)=un+1(r,t)-un(r,t),  en(r,t)=un(r,t)-u(r,t), |en(r,t*)|=maxt|en(r,t)| then since en is a decreasing function with respect to t from the mean value theorem we can write,
(3.11)en+1(r,t)=en(r,t)+Lt-1(-en(r,t)+∫0t[D2(un(x,τ))-D2(u(x,τ))]dτ-∫0t[F(un(x,τ))-F(u(x,τ))]dτ)≤en(r,t)+Lt-1[-en(r,t)+Lt-1|en(r,t)|(ν(L1+TL2))]≤en(r,t)-Ten(r,η)+T(M′L1+M′L2)Lt-1Lt-1|en(r,t)|≤(1-T(1-α))|en(r,t*)|,
where 0≤η≤t. Hence, en+1(r,t)≤β|en(r,t*)|.
Therefore,(3.12)‖en+1‖=max⁡∀t∈J⁡|en+1|≤βmax⁡∀t∈J⁡|en|≤β‖en‖.
Since 0<β<1, then ∥en∥→0. So, the series converges and the proof is complete.Theorem 3.4.
If the series solution (2.38) of problem (1.1) is convergent then it converges to the exact solution of the problem (1.1) by using HAM.Proof.
We assume:(3.13)u(x,t)=∑m=0∞um(x,t)
where
(3.14)lim⁡m→∞um(x,t)=0.
We can write,(3.15)∑m=1n[um(x,t)-χmum-1(x,t)]=u1+(u2-u1)+⋯+(un-un-1)=un(x,t).
We have,(3.16)lim⁡n→∞un(x,t)=0.
So, using (3.16) and the definition of the nonlinear operator L, we have(3.17)∑m=1∞L[um(x,t)-χmum-1(x,t)]=L[∑m=1∞[um(x,t)-χmum-1(x,t)]]=0.
Therefore from (2.32), we can obtain that,
(3.18)∑m=1∞L[um(x,t)-χmum-1(x,t)]=hH(x,t)∑m=1∞Rm-1(um-1(x,t))=0.
Sinceh≠0 and H(x,t)≠0, we have(3.19)∑m=1∞Rm-1(um-1(x,t))=0.
By substitutingℜm-1(um-1(x,t)) into the relation (3.19) and simplifying it, we have(3.20)∑m=1∞Rm-1(um-1(x,t))=∑m=1∞[um-1(x,t)-∫0tD2(um-1(x,τ))dτ-∫0tF(um-1(x,τ))dτ-(1-χm)f(x)]=u(x,t)-f(x)-∫0tD2(um-1(x,τ))dτ-∫0t[F(um-1(x,τ))dτ].
From (3.19) and (3.20), we have(3.21)u(x,t)=G(x,t)+∫0t(t-τ)D2(u(x,τ))dτ-∫0t(t-τ)F(u(x,τ))dτ,
therefore, u(x,t) must be the exact solution of (1.1).Theorem 3.5.
If|um(x,t)|≤1, then the series solution (2.46) of problem (1.1) converges to the exact solution.Proof.
We can write the solutionu(x,t) as follows:
(3.22)u(x,t)=∑m=0∞um(x,t)=∑m=0∞{∫0tD2(um-1(x,τ))dτ+∫0tF(um-1(x,τ))dτ+(1-χm)f(x)+∫0t(t-τ)g1(um-1(x))dτ+∫0t(t-τ)(g2(x))dτ}.
If|um(x,t)|<1, therefore, u(x,t)=∑m=0∞um(x,t) must be the exact solution of (1.1).


## 4. Numerical Example

In this section, we compute a numerical example which is solved by the MADM, VIM, HAM and MHPM. The program has been provided with Mathematica 6 according to the following algorithm. In this algorithmɛ is a given positive value.Algorithm 4.1.
One has the following.Step 1.
Setn←0.Step 2.
Calculate the recursive relation (2.10) for MADM, (2.20) for VIM, (2.38) for HAM and (2.46) for MHPM.Step 3.
If|un+1-un|<ɛ then go to Step 4, else n←n+1 and go to Step 2.Step 4.
Printu(x,t)=∑i=0nui(x,t) as the approximate of the exact solution.Example 4.2 (see [3]).
Consider the Fisher equation withs=3. 							(4.1)ut=uxx+u(1-u3),
subject to initial conditions
(4.2)u(x,0)=(1(1+e(3/10)x)1/3)2,
with the exact solution is {(1/2)tanh[-(3/210)(x-(7/10)t)]+(1/2)}2/3.


## 5. Conclusion

The HAM 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 paper, the HAM has been successfully employed to obtain the approximate analytical solution of the Fisher equation. For this purpose, we showed that the HAM is more rapid convergence than the MADM, VIM and MHPM.Table 1
Numerical results of Example4.2.
tErrorErrorErrorError(MADM,n=5)(VIM,n=3)(HAM,n=2)(MHPM,n=3)0.50.05883020.05871110.03482650.05886680.60.06700540.06778650.04377780.06746310.70.0685590.06747420.04241780.06700320.80.07453420.07203830.05587520.07204280.90.07453420.07463310.05662340.074589431.00.07663310.07750120.05997350.0775367



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*Source: 254675-2010-12-13.xml*