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International Journal of Differential Equations
Volumeย 2010, Article IDย 254675, 16 pages
http://dx.doi.org/10.1155/2010/254675
Research Article

Convergence of Iterative Methods Applied to Generalized Fisher Equation

Young Researchers Club, Islamic Azad University, Central Tehran Branch, P.O. Box 15655/461, Tehran, Iran

Received 12 April 2010; Accepted 13 September 2010

Academic Editor: Christo I.ย Christov

Copyright ยฉ 2010 Sh. Sadigh Behzadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Fisher proposed equation ๐œ•๐‘ข/๐œ•๐‘ก=๐œ•2๐‘ข/๐œ•๐‘ฅ2+๐‘ข(1โˆ’๐‘ข) as a model for the propagation of a mutant gene, with ๐‘ข 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 [3โ€“5], an analytical study of Fisher equation by using Adomian decomposition method [6], numerical solution for solving Burger-Fisher equation [7โ€“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: ๐œ•๐‘ข=๐œ•๐œ•๐‘ก2๐‘ข๐œ•๐‘ฅ2+๐‘ข(1โˆ’๐‘ข๐‘ ),(1.1) with the initial conditions given by๐‘ข(๐‘ฅ,0)=๐‘“(๐‘ฅ).(1.2)

The paper is organized as follows. In Section 2, 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 ๐‘ก and using the initial conditions, we obtain๎€œ๐‘ข(๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ)+๐‘ก0๐œ•2๐‘ข(๐‘ฅ,๐œ)๐œ•๐‘ฅ2๎€œ๐‘‘๐œ+๐‘ก0๐‘ข(๐‘ฅ,๐œ)(1โˆ’๐‘ข๐‘ (๐‘ฅ,๐œ))๐‘‘๐œ.(1.3)

We set ๐น(๐‘ข)=๐‘ข(1โˆ’๐‘ข๐‘ ).(1.4)

In (1.3), we assume ๐‘“(๐‘ฅ) is bounded for all ๐‘ฅ in ๐ฝ=[0,๐‘‡](๐‘‡โˆˆโ„) and|๐‘กโˆ’๐œ|โ‰ค๐‘€๎…ž,โˆ€0โ‰ค๐‘ก,๐œโ‰ค๐‘‡.(1.5)

The terms ๐ท2(๐‘ข) and ๐น(๐‘ข) are Lipschitz continuous with |๐ท2(๐‘ข)โˆ’๐ท2(๐‘ขโˆ—)|โ‰ค๐ฟ1|๐‘ขโˆ’๐‘ขโˆ—|, |๐น(๐‘ข)โˆ’๐น(๐‘ขโˆ—)|โ‰ค๐ฟ2|๐‘ขโˆ’๐‘ขโˆ—|.

We set๎€ท๐‘€๐›ผ=๐‘‡๎…ž๐ฟ1+๐‘€๎…ž๐ฟ2๎€ธ,๐›ฝ=1โˆ’๐‘‡(1โˆ’๐›ผ).(1.6)

Now we decompose the unknown function ๐‘ข(๐‘ฅ,๐‘ก) by a sum of components defined by the following decomposition series with ๐‘ข0 identified as ๐‘ข(๐‘ฅ,0):๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘›=0๐‘ข๐‘›(๐‘ฅ,๐‘ก).(1.7)

2. Iterative Methods

2.1. Preliminaries of the MADM

The Adomian decomposition method is applied to the following general nonlinear equation:๐ฟ๐‘ข+๐‘…๐‘ข+๐‘๐‘ข=๐‘”(๐‘ฅ),(2.1) where ๐‘ข is the unknown function, ๐ฟ is the highest-order derivative which is assumed to be easily invertible, ๐‘… is a linear differential operator of order less than ๐ฟ,๐‘๐‘ข represents the nonlinear terms, and ๐‘” is the source term. Applying the inverse operator ๐ฟโˆ’1 to both sides of (2.1), and using the given conditions, we obtain ๐‘ข=๐‘“(๐‘ฅ)โˆ’๐ฟโˆ’1(๐‘…๐‘ข)โˆ’๐ฟโˆ’1(๐‘๐‘ข),(2.2) where the function ๐‘“(๐‘ฅ) represents the terms arising from integrating the source term ๐‘”(๐‘ฅ). The nonlinear operator ๐‘๐‘ข=๐บ(๐‘ข) is decomposed as๐บ(๐‘ข)=โˆž๎“๐‘›=0๐ด๐‘›,(2.3) where ๐ด๐‘›, ๐‘›โ‰ฅ0 are the Adomian polynomials determined formally as follows:๐ด๐‘›=1๎ƒฌ๐‘‘๐‘›!๐‘›๐‘‘๐œ†๐‘›๎ƒฌ๐‘๎ƒฉโˆž๎“๐‘–=0๐œ†๐‘–๐‘ข๐‘–๎ƒช๎ƒญ๎ƒญ๐œ†=0.(2.4) Adomian polynomials were introduced in [12โ€“15] as๐ด0๎€ท๐‘ข=๐บ0๎€ธ,๐ด1=๐‘ข1๐บ๎…ž๎€ท๐‘ข0๎€ธ,๐ด2=๐‘ข2๐บ๎…ž๎€ท๐‘ข0๎€ธ+1๐‘ข2!21๐บ๎…ž๎…ž๎€ท๐‘ข0๎€ธ,๐ด3=๐‘ข3๐บ๎…ž๎€ท๐‘ข0๎€ธ+๐‘ข1๐‘ข2๐บ๎…ž๎…ž๎€ท๐‘ข0๎€ธ+1๐‘ข3!31๐บ๎…ž๎…ž๎…ž๎€ท๐‘ข0๎€ธ,โ€ฆ.(2.5)

2.1.1. Adomian Decomposition Method

The standard decomposition technique represents the solution of ๐‘ข in (2.1) as the following series,๐‘ข=โˆž๎“๐‘›=0๐‘ข๐‘–,(2.6) where, the components ๐‘ข0,๐‘ข1,โ€ฆ are usually determined recursively by๐‘ข0๐‘ข=๐‘“(๐‘ฅ),๐‘›+1=โˆ’๐ฟโˆ’1๎€ท๐‘…๐‘ข๐‘›๎€ธโˆ’๐ฟโˆ’1๎€ท๐ด๐‘›๎€ธ,๐‘›โ‰ฅ0.(2.7) Substituting (2.5) into (2.7) leads to the determination of the components of ๐‘ข. Having determined the components ๐‘ข0,๐‘ข1,โ€ฆ the solution ๐‘ข 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 ๐‘“(๐‘ฅ) can be divided into two parts, namely ๐‘“1(๐‘ฅ) and ๐‘“2(๐‘ฅ). Under this assumption we set๐‘“(๐‘ฅ)=๐‘“1(๐‘ฅ)+๐‘“2(๐‘ฅ).(2.8) Accordingly, a slight variation was proposed only on the components ๐‘ข0 and ๐‘ข1. The suggestion was that only the part ๐‘“1 be assigned to the zeroth component ๐‘ข0, whereas the remaining part ๐‘“2 be combined with the other terms given in (2.7) to define ๐‘ข1. Consequently, the modified recursive relation๐‘ข0=๐‘“1๐‘ข(๐‘ฅ),1=๐‘“2(๐‘ฅ)โˆ’๐ฟโˆ’1๎€ท๐‘…๐‘ข0๎€ธโˆ’๐ฟโˆ’1๎€ท๐ด0๎€ธ,โ‹ฎ๐‘ข๐‘›+1=โˆ’๐ฟโˆ’1๎€ท๐‘…๐‘ข๐‘›๎€ธโˆ’๐ฟโˆ’1๎€ท๐ด๐‘›๎€ธ,๐‘›โ‰ฅ1,(2.9) 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:๐‘ข0(๐‘ฅ,๐‘ก)=๐‘“1๐‘ข(๐‘ฅ),1(๐‘ฅ,๐‘ก)=๐‘“2(๎€œ๐‘ฅ)+๐‘ก0๐ท2๎€ท๐‘ข0(๎€ธ+๎€œ๐‘ฅ,๐œ)๐‘ก0๐น๎€ท๐‘ข0(๎€ธโ‹ฎ๐‘ข๐‘ฅ,๐œ)๐‘‘๐œ,๐‘›+1๎€œ(๐‘ฅ,๐‘ก)=๐‘ก0๐ท2๎€ท๐‘ข๐‘›๎€ธ+๎€œ(๐‘ฅ,๐œ)๐‘ก0๐น๎€ท๐‘ข๐‘›๎€ธ(๐‘ฅ,๐œ)๐‘‘๐œ.(2.10)

The operators ๐ท2(๐‘ข(๐‘ฅ,๐œ))=(๐‘‘2/๐‘‘๐‘ฅ2)๐‘ข(๐‘ฅ,๐‘ก) and ๐น(๐‘ข(๐‘ฅ,๐œ)) are usually represented by an infinite series of the so-called Adomian polynomials as follows:๐น(๐‘ข)=โˆž๎“๐‘–=0๐ด๐‘–,๐ท2(๐‘ข)=โˆž๎“๐‘–=0๐ฟ๐‘–.(2.11) where ๐ด๐‘– and ๐ฟ๐‘–(๐‘–โ‰ฅ0) are the Adomian polynomials were introduced in [12].

From [12], we can write another formula for the Adomian polynomials:๐ฟ๐‘›=๐ท2๎€ท๐‘ ๐‘›๎€ธโˆ’๐‘›โˆ’1๎“๐‘–=0๐ฟ๐‘–,๐ด๐‘›๎€ท๐‘ =๐น๐‘›๎€ธโˆ’๐‘›โˆ’1๎“๐‘–=0๐ด๐‘–,(2.12) where the partial sum is ๐‘ ๐‘›=โˆ‘๐‘›๐‘–=0๐‘ข๐‘–(๐‘ฅ,๐‘ก).

2.3. Preliminaries of the VIM

In the VIM [17โ€“20], we consider the following nonlinear differential equation:๐ฟ(๐‘ข)+๐‘(๐‘ข)=๐‘”(๐‘ก),(2.13) where ๐ฟ is a linear operator,๐‘ is a nonlinear operator and ๐‘”(๐‘ฅ,๐‘ก) is a known analytical function. In this case, a correction functional can be constructed as follows:๐‘ข๐‘›+1(๐‘ฅ,๐‘ก)=๐‘ข๐‘›(๎€œ๐‘ฅ,๐‘ก)+๐‘ก0๎€ฝ๐ฟ๎€ท๐‘ข๐œ†(๐‘ฅ,๐œ)๐‘›(๎€ธ๎€ท๐‘ข๐‘ฅ,๐œ)+๐‘๐‘›(๎€ธ๎€พ๐‘ฅ,๐œ)โˆ’๐‘”(๐‘ฅ,๐œ)๐‘‘๐œ,๐‘›โ‰ฅ0,(2.14) where ๐œ† is a general Lagrange multiplier which can be identified optimally via variational theory. Here the function ๐‘ข๐‘›(๐‘ฅ,๐œ) is a restricted variations which means ๐›ฟ๐‘ข๐‘›=0. Therefore, we first determine the Lagrange multiplier ๐œ† that will be identified optimally via integration by parts. The successive approximation ๐‘ข๐‘›(๐‘ฅ,๐‘ก), ๐‘›โ‰ฅ0 of the solution ๐‘ข(๐‘ก) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function ๐‘ข0. The zeroth approximation ๐‘ข0 may be selected any function that just satisfies at least the initial and boundary conditions. With ๐œ† determined, then several approximation ๐‘ข๐‘›(๐‘ฅ,๐‘ก), ๐‘›โ‰ฅ0 follow immediately. Consequently, the exact solution may be obtained by using ๐‘ข(๐‘ฅ,๐‘ก)=lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘ฅ,๐‘ก).(2.15)

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: ๐‘ข๐‘›+1(๐‘ฅ,๐‘ก)=๐‘ข๐‘›(๐‘ฅ,๐‘ก)+๐ฟ๐‘กโˆ’1๎‚ต๐œ†๎‚ธ๎€œ๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘›(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘›(๎€ธ,๐‘ฅ,๐œ)๐‘‘๐œ๎‚น๎‚ถ(2.16) where, ๐ฟ๐‘กโˆ’1(๎€œโ‹…)=๐‘ก0(โ‹…)๐‘‘๐œ.(2.17)

To find the optimal ๐œ†, we proceed as๐›ฟ๐‘ข๐‘›+1(๐‘ฅ,๐‘ก)=๐›ฟ๐‘ข๐‘›(๐‘ฅ,๐‘ก)+๐›ฟ๐ฟ๐‘กโˆ’1๎‚ต๐œ†๎‚ธ๐‘ข๐‘›(๎€œ๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘›(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0๐น๎€ท๐‘ข๐‘›(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ๎‚น๎‚ถ=๐›ฟ๐‘ข๐‘›(๐‘ฅ,๐‘ก)+๐œ†(๐‘ฅ)๐›ฟ๐‘ข๐‘›(๐‘ฅ,๐‘ก)โˆ’๐ฟ๐‘กโˆ’1๎€บ๐›ฟ๐‘ข๐‘›(๐‘ฅ,๐‘ก)๐œ†๎…ž๎€ป.(๐‘ฅ)(2.18)

From (2.18), the stationary conditions can be obtained as follows:๐œ†๎…ž=0,1+๐œ†=0.(2.19)

Therefore, the Lagrange multipliers can be identified as ๐œ†=โˆ’1 and by substituting in (2.16), the following iteration formula is obtained.๐‘ข0๐‘ข(๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ),๐‘›+1(๐‘ฅ,๐‘ก)=๐‘ข๐‘›(๐‘ฅ,๐‘ก)โˆ’๐ฟ๐‘กโˆ’1๐‘ข๎‚ต๎‚ธ๐‘›(๎€œ๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘›(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘›(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ๎‚น๎‚ถ,๐‘›โ‰ฅ0.(2.20)

Relation (2.20) will enable us to determine the components ๐‘ข๐‘›(๐‘ฅ,๐‘ก) recursively for ๐‘›โ‰ฅ0.

2.5. Preliminaries of the HAM

Consider๐‘[๐‘ข]=0,(2.21) where ๐‘ is a nonlinear operator, ๐‘ข(๐‘ฅ,๐‘ก) is unknown function and ๐‘ฅ is an independent variable. let ๐‘ข0(๐‘ฅ,๐‘ก) denote an initial guess of the exact solution ๐‘ข(๐‘ฅ,๐‘ก), โ„Žโ‰ 0 an auxiliary parameter, ๐ป(๐‘ฅ,๐‘ก)โ‰ 0 an auxiliary function, and ๐ฟ an auxiliary nonlinear operator with the property ๐ฟ[๐‘Ÿ(๐‘ฅ,๐‘ก)]=0 when ๐‘Ÿ(๐‘ฅ,๐‘ก)=0. Then using ๐‘žโˆˆ[0,1] as an embedding parameter, we construct a homotopy as follows:๎€บ(1โˆ’๐‘ž)๐ฟ๐œ™(๐‘ฅ,๐‘ก;๐‘ž)โˆ’๐‘ข0๎€ป[]=๎๐ป๎€บ(๐‘ฅ,๐‘ก)โˆ’๐‘žโ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘๐œ™(๐‘ฅ,๐‘ก;๐‘ž)๐œ™(๐‘ฅ,๐‘ก;๐‘ž);๐‘ข0๎€ป.(๐‘ฅ,๐‘ก),๐ป(๐‘ฅ,๐‘ก),โ„Ž,๐‘ž(2.22)

It should be emphasized that we have great freedom to choose the initial guess ๐‘ข0(๐‘ฅ,๐‘ก), the auxiliary nonlinear operator ๐ฟ, the nonzero auxiliary parameter โ„Ž, and the auxiliary function ๐ป(๐‘ฅ,๐‘ก).

Enforcing the homotopy (2.22) to be zero, that is, ๎๐ป๎€บ๐œ™(๐‘ฅ,๐‘ก;๐‘ž);๐‘ข0๎€ป(๐‘ฅ,๐‘ก),๐ป(๐‘ฅ,๐‘ก),โ„Ž,๐‘ž=0,(2.23)

we have the so-called zero-order deformation equation๎€บ๐œ™(1โˆ’๐‘ž)๐ฟ(๐‘ฅ,๐‘ก;๐‘ž)โˆ’๐‘ข0๎€ป[๐œ™](๐‘ฅ,๐‘ก)=๐‘žโ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘(๐‘ฅ,๐‘ก;๐‘ž).(2.24)

When ๐‘ž=0, the zero-order deformation (2.24) becomes๐œ™(๐‘ฅ,๐‘ก;0)=๐‘ข0(๐‘ฅ,๐‘ก),(2.25) and when ๐‘ž=1, since โ„Žโ‰ 0 and ๐ป(๐‘ฅ,๐‘ก)โ‰ 0, the zero-order deformation (2.24) is equivalent to๐œ™(๐‘ฅ,๐‘ก;1)=๐‘ข(๐‘ฅ,๐‘ก).(2.26)

Thus, according to (2.25) and (2.26), as the embedding parameter ๐‘ž increases from 0 to 1, ๐œ™(๐‘ฅ,๐‘ก;๐‘ž) varies continuously from the initial approximation ๐‘ข0(๐‘ฅ,๐‘ก) to the exact solution ๐‘ข(๐‘ฅ,๐‘ก). Such a kind of continuous variation is called deformation in homotopy [21, 22].

Due to Taylor's theorem, ๐œ™(๐‘ฅ,๐‘ก;๐‘ž) can be expanded in a power series of ๐‘ž as follows:๐œ™(๐‘ฅ,๐‘ก;๐‘ž)=๐‘ข0(๐‘ฅ,๐‘ก)+โˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š,(2.27) where๐‘ข๐‘š1(๐‘ฅ,๐‘ก)=๐œ•๐‘š!๐‘š๐œ™(๐‘ฅ,๐‘ก;๐‘ž)๐œ•๐‘ž๐‘š||||๐‘ž=0.(2.28)

Let the initial guess ๐‘ข0(๐‘ฅ,๐‘ก), the auxiliary nonlinear parameter ๐ฟ, the nonzero auxiliary parameter โ„Ž and the auxiliary function ๐ป(๐‘ฅ,๐‘ก) be properly chosen so that the power series (2.27) of ๐œ™(๐‘ฅ,๐‘ก;๐‘ž) converges at ๐‘ž=1, then, we have under these assumptions the solution series๐‘ข(๐‘ฅ,๐‘ก)=๐œ™(๐‘ฅ,๐‘ก;1)=๐‘ข0(๐‘ฅ,๐‘ก)+โˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก).(2.29)

From (2.27), we can write (2.24) as follows:๎€บ๐œ™(1โˆ’๐‘ž)๐ฟ(๐‘ฅ,๐‘ก;๐‘ž)โˆ’๐‘ข0๎€ป๎ƒฌ(๐‘ฅ,๐‘ก)=(1โˆ’๐‘ž)๐ฟโˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š๎ƒญ[]๎ƒฌ=๐‘žโ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘๐œ™(๐‘ฅ,๐‘ก;๐‘ž)โŸน๐ฟโˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š๎ƒญ๎ƒฌโˆ’๐‘ž๐ฟโˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š๎ƒญ[].=๐‘žโ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘๐œ™(๐‘ฅ,๐‘ก;๐‘ž)(2.30)

By differentiating (2.30) ๐‘š times with respect to ๐‘ž, we obtain๎ƒฏ๐ฟ๎ƒฌโˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š๎ƒญ๎ƒฌโˆ’๐‘ž๐ฟโˆž๎“๐‘š=1๐‘ข๐‘š(๐‘ฅ,๐‘ก)๐‘ž๐‘š๎ƒญ๎ƒฐ(๐‘š)[]}={๐‘žโ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘๐œ™(๐‘ฅ,๐‘ก;๐‘ž)(๐‘š)๎€บ๐‘ข=๐‘š!๐ฟ๐‘š(๐‘ฅ,๐‘ก)โˆ’๐‘ข๐‘šโˆ’1๎€ป๐œ•(๐‘ฅ,๐‘ก)=โ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘š๐‘šโˆ’1๐‘[๐œ™(๐‘ฅ,๐‘ก;๐‘ž)]๐œ•๐‘ž๐‘šโˆ’1||||๐‘ž=0.(2.31)

Therefore,๐ฟ๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1๎€ป(๐‘ฅ,๐‘ก)=โ„Ž๐ป(๐‘ฅ,๐‘ก)โ„œ๐‘š๎€ท๐‘ฆ๐‘šโˆ’1๎€ธ(๐‘ฅ),(2.32) where,โ„œ๐‘š๎€ท๐‘ข๐‘šโˆ’1๎€ธ=1(๐‘ฅ,๐‘ก)๐œ•(๐‘šโˆ’1)!๐‘šโˆ’1๐‘[๐œ™(๐‘ฅ,๐‘ก;๐‘ž)]๐œ•๐‘ž๐‘šโˆ’1||||๐‘ž=0,๐œ’(2.33)๐‘š=๎ƒฏ0,๐‘šโ‰ค1,1,๐‘š>1.(2.34)

Note that the high-order deformation (2.32) is governing the nonlinear operator ๐ฟ, and the term โ„œ๐‘š(๐‘ข๐‘šโˆ’1(๐‘ฅ,๐‘ก)) can be expressed simply by (2.33) for any nonlinear operator ๐‘.

2.6. Description of the HAM

To obtain the approximation solution of (1.1), according to HAM, let๐‘[๐‘ข]๎€œ=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2(๎€œ๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œโˆ’๐‘ก0๐น(๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ,(2.35) so โ„œ๐‘š๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๐‘ฅ,๐‘ก)=๐‘ข๐‘šโˆ’1(๎€œ๐‘ฅ,๐‘ก)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๎€ท๐‘ฅ,๐œ)๐‘‘๐œโˆ’1โˆ’๐œ’๐‘š๎€ธ๐‘“(๐‘ฅ).(2.36)

Substituting (2.36) into(2.32)๐ฟ๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1๎€ป๎‚ธ๐‘ข(๐‘ฅ,๐‘ก)=โ„Ž๐ป(๐‘ฅ,๐‘ก)๐‘šโˆ’1๎€œ(๐‘ฅ,๐‘ก)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€œ(๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€ท(๐‘ฅ,๐œ)๐‘‘๐œโˆ’1โˆ’๐œ’๐‘š๎€ธ๐‘“๎‚น.(๐‘ฅ)(2.37)

We take an initial guess ๐‘ข0(๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ), an auxiliary nonlinear operator ๐ฟ๐‘ข=๐‘ข, a nonzero auxiliary parameter โ„Ž=โˆ’1, and auxiliary function ๐ป(๐‘ฅ,๐‘ก)=1. This is substituted into (2.37) to give the recurrence relation ๐‘ข0๐‘ข(๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ),๐‘›(๎€œ๐‘ฅ,๐‘ก)=๐‘ก0๐ท2๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0๐น๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ,๐‘›โ‰ฅ1.(2.38)

Therefore, the solution ๐‘ข(๐‘ฅ,๐‘ก) becomes๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘›=0๐‘ข๐‘›(๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ)+โˆž๎“๐‘›=0๎‚ต๎€œ๐‘ก0๐ท2๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0๐น๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๎‚ถ๐‘ฅ,๐œ)๐‘‘๐œ,(2.39) which is the method of successive approximations. If||๐‘ข๐‘›||(๐‘ฅ,๐‘ก)<1(2.40) then the series solution (2.39) convergence uniformly.

2.7. Description of the MHPM

To explain MHPM, we consider (1.1) as๎€œ๐ฟ(๐‘ข)=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘›โˆ’1(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ,(2.41) where ๐ท2(๐‘ข(๐‘ฅ,๐œ))=๐‘”1(๐‘ฅ)โ„Ž1(๐œ) and ๐น(๐‘ข(๐‘ฅ,๐œ))=๐‘”2(๐‘ฅ)โ„Ž2(๐œ). We can define homotopy ๐ป(๐‘ข,๐‘,๐‘š) by๐ป(๐‘ข,๐‘œ,๐‘š)=๐‘“(๐‘ข),๐ป(๐‘ข,1,๐‘š)=๐ฟ(๐‘ข),(2.42) where ๐‘š is an unknown real number and๐‘“(๐‘ข(๐‘ฅ,๐‘ก))=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐บ(๐‘ฅ,๐‘ก).(2.43) Typically we may choose a convex homotopy by๐ป๎€บ๐‘š๎€ท๐‘”(๐‘ข,๐‘,๐‘š)=(1โˆ’๐‘)๐‘“(๐‘ข)+๐‘๐ฟ(๐‘ข)+๐‘(1โˆ’๐‘)1(๐‘ฅ)+๐‘”2(๐‘ฅ)๎€ธ๎€ป=0,0โ‰ค๐‘โ‰ค1.(2.44) where ๐‘š is called the accelerating parameters, and for ๐‘š=0 we define ๐ป(๐‘ข,๐‘,0)=๐ป(๐‘ข,๐‘), which is the standard HPM. The convex homotopy (2.44) continuously trace an implicity defined curve from a starting point ๐ป(๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ข),0,๐‘š) to a solution function ๐ป(๐‘ข(๐‘ฅ,๐‘ก),1,๐‘š). The embedding parameter ๐‘ monotonically increase from o to 1 as trivial problem ๐‘“(๐‘ข)=0 is continuously deformed to original problem ๐ฟ(๐‘ข)=0. [23, 24]

The MHPM uses the homotopy parameter ๐‘ as an expanding parameter to obtain๐‘ฃ=โˆž๎“๐‘›=0๐‘๐‘›๐‘ข๐‘›;(2.45) when ๐‘โ†’1 (2.44) corresponds to the original one, (2.45) becomes the approximate solution of (1.1), that is,๐‘ข=lim๐‘โ†’1๐‘ฃ=โˆž๎“๐‘š=0๐‘ข๐‘š,(2.46) where,๐‘ข๐‘š(๎€œ๐‘ฅ,๐‘ก)=๐‘“(๐‘ฅ)+๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0๐น๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ.(2.47)

3. Existence and Convergency of Iterative Methods

Theorem 3.1. Let 0<๐›ผ<1, then Fisher equation (1.1), has a unique solution.

Proof. Let ๐‘ข and ๐‘ขโˆ— be two different solutions of (1.3) then ||๐‘ขโˆ’๐‘ขโˆ—||=||||๎€œ๐‘ก0๎€บ๐ท2(๐‘ข(๐‘ฅ,๐œ))โˆ’๐ท2๎€ท๐‘ขโˆ—๎€œ(๐‘ฅ,๐œ)๎€ธ๎€ป๐‘‘๐œ+๐‘ก0๎€บ๎€ท๐‘ข๐น(๐‘ข(๐‘ฅ,๐œ))โˆ’๐นโˆ—||||โ‰ค๎€œ(๐‘ฅ,๐œ)๎€ธ๎€ป๐‘‘๐œ๐‘ก0||๐ท2(๐‘ข(๐‘ฅ,๐œ))โˆ’๐ท2๎€ท๐‘ขโˆ—๎€ธ||๎€œ(๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0||๐น๎€ท๐‘ข(๐‘ข(๐‘ฅ,๐œ))โˆ’๐นโˆ—๎€ธ||๎€ท๐‘€(๐‘ฅ,๐œ)๐‘‘๐œโ‰ค๐‘‡๎…ž๐ฟ1+๐‘€๎…ž๐ฟ2๎€ธ||๐‘ขโˆ’๐‘ขโˆ—||||=๐›ผ๐‘ขโˆ’๐‘ขโˆ—||.(3.1)
From which we get (1โˆ’๐›ผ)|๐‘ขโˆ’๐‘ขโˆ—|โ‰ค0. Since 0<๐›ผ<1. then |๐‘ขโˆ’๐‘ขโˆ—|=0. Implies ๐‘ข=๐‘ขโˆ— and completes the proof.

Theorem 3.2. The series solution โˆ‘๐‘ข(๐‘ฅ,๐‘ก)=โˆž๐‘–=0๐‘ข๐‘–(๐‘ฅ,๐‘ก) of problem (1.1) using MADM convergence when 0<๐›ผ<1, |๐‘ข1(๐‘ฅ,๐‘ก)|<โˆž.

Proof. Denote as (๐ถ[๐ฝ],โ€–โ‹…โ€–) the Banach space of all continuous functions on ๐ฝ with the norm โ€–๐‘“(๐‘ก)โ€–=max|๐‘“(๐‘ก)|, for all ๐‘ก in ๐ฝ. Define the sequence of partial sums ๐‘ ๐‘›, and let ๐‘ ๐‘› and ๐‘ ๐‘š be arbitrary partial sums with ๐‘›โ‰ฅ๐‘š. We are going to prove that ๐‘ ๐‘› is a Cauchy sequence in this Banach space: โ€–โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–=maxโˆ€๐‘กโˆˆ๐ฝ||๐‘ ๐‘›โˆ’๐‘ ๐‘š||=maxโˆ€๐‘กโˆˆ๐ฝ|||||๐‘›๎“๐‘–=๐‘š+1๐‘ข๐‘–|||||(๐‘ฅ,๐‘ก)=maxโˆ€๐‘กโˆˆ๐ฝ|||||๐‘›๎“๐‘–=๐‘š+1๎€œ๐‘ก0๐ฟ๐‘–โˆ’1๐‘‘๐œ+๐‘›๎“๐‘–=๐‘š+1๎€œ๐‘ก0๐ด๐‘–โˆ’1|||||๐‘‘๐œ=maxโˆ€๐‘กโˆˆ๐ฝ|||||๎€œ๐‘ก0๎ƒฉ๐‘›โˆ’1๎“๐‘–=๐‘š๐ฟ๐‘–๎ƒช๎€œ๐‘‘๐œ+๐‘ก0๎ƒฉ๐‘›โˆ’1๎“๐‘–=๐‘š๐ด๐‘–๎ƒช|||||.๐‘‘๐œ(3.2)
From [12], we have๐‘›โˆ’1๎“๐‘–=๐‘š๐ฟ๐‘–=๐ท2๎€ท๐‘ ๐‘›โˆ’1๎€ธโˆ’๐ท2๎€ท๐‘ ๐‘šโˆ’1๎€ธ,๐‘›โˆ’1๎“๐‘–=๐‘š๐ด๐‘–๎€ท๐‘ =๐น๐‘›โˆ’1๎€ธ๎€ท๐‘ โˆ’๐น๐‘šโˆ’1๎€ธ.(3.3)
So,โ€–โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–=maxโˆ€๐‘กโˆˆ๐ฝ||||๎€œ๐‘ก0๎€บ๐ท2๎€ท๐‘ ๐‘›โˆ’1๎€ธโˆ’๐ท2๎€ท๐‘ ๐‘šโˆ’1๎€œ๎€ธ๎€ป๐‘‘๐œ+๐‘ก0๎€บ๐น๎€ท๐‘ ๐‘›โˆ’1๎€ธ๎€ท๐‘ โˆ’๐น๐‘šโˆ’1||||โ‰ค๎€œ๎€ธ๎€ป๐‘‘๐œ๐‘ก0||๐ท2๎€ท๐‘ ๐‘›โˆ’1๎€ธโˆ’๐ท2๎€ท๐‘ ๐‘šโˆ’1๎€ธ||๎€œ๐‘‘๐œ+๐‘ก0||๐น๎€ท๐‘ ๐‘›โˆ’1๎€ธ๎€ท๐‘ โˆ’๐น๐‘šโˆ’1๎€ธ||โ€–โ€–๐‘ ๐‘‘๐œโ‰ค๐›ผ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–.(3.4)
Let ๐‘›=๐‘š+1, thenโ€–โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–โ€–โ€–๐‘ โ‰ค๐›ผ๐‘šโˆ’๐‘ ๐‘šโˆ’1โ€–โ€–โ‰ค๐›ผ2โ€–โ€–๐‘ ๐‘šโˆ’1โˆ’๐‘ ๐‘šโˆ’2โ€–โ€–โ‰คโ‹ฏโ‰ค๐›ผ๐‘šโ€–โ€–๐‘ 1โˆ’๐‘ 0โ€–โ€–.(3.5)
From the triangle inquality we haveโ€–โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–โ‰คโ€–โ€–๐‘ ๐‘š+1โˆ’๐‘ ๐‘šโ€–โ€–+โ€–โ€–๐‘ ๐‘š+2โˆ’๐‘ ๐‘š+1โ€–โ€–โ€–โ€–๐‘ +โ‹ฏ+๐‘›โˆ’๐‘ ๐‘›โˆ’1โ€–โ€–โ‰ค๎€บ๐›ผ๐‘š+๐›ผ๐‘š1+โ‹ฏ+๐›ผ๐‘›โˆ’๐‘šโˆ’1๎€ปโ€–โ€–๐‘ 1โˆ’๐‘ 0โ€–โ€–โ‰ค๐›ผ๐‘š๎€บ1+๐›ผ+๐›ผ2+โ‹ฏ+๐›ผ๐‘›โˆ’๐‘šโˆ’1๎€ปโ€–โ€–๐‘ 1โˆ’๐‘ 0โ€–โ€–โ‰ค๎‚ธ1โˆ’๐›ผ๐‘›โˆ’๐‘š๎‚นโ€–โ€–๐‘ข1โˆ’๐›ผ1โ€–โ€–.(๐‘ฅ,๐‘ก)(3.6)
Since 0<๐›ผ<1, we have (1โˆ’๐›ผ๐‘›โˆ’๐‘š)<1, thenโ€–โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ€–โ‰ค๐›ผ๐‘š1โˆ’๐›ผmaxโˆ€๐‘กโˆˆ๐ฝ||๐‘ข1||.(๐‘ฅ,๐‘ก)(3.7)
But |๐‘ข1(๐‘ฅ,๐‘ก)|<โˆž, so, as ๐‘šโ†’โˆž, then โ€–๐‘ ๐‘›โˆ’๐‘ ๐‘šโ€–โ†’0. We conclude that ๐‘ ๐‘› is a Cauchy sequence in ๐ถ[๐ฝ], therefore the series is convergence and the proof is complete.

Theorem 3.3. The series solution โˆ‘๐‘ข(๐‘ฅ,๐‘ก)=โˆž๐‘–=0๐‘ข๐‘–(๐‘ฅ,๐‘ก) of problem (1.1) using VIM converges when 0<๐›ผ<1, 0<๐›ฝ<1.

Proof. One has the following: ๐‘ข๐‘›+1(๐‘ฅ,๐‘ก)=๐‘ข๐‘›(๐‘ฅ,๐‘ก)โˆ’๐ฟ๐‘กโˆ’1๐‘ข๎‚ต๎‚ธ๐‘›(๎€œ๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘›(๎€ธ๎€œ๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘›(๎€ธ๐‘ฅ,๐œ)๐‘‘๐œ๎‚น๎‚ถ,(3.8)๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐ฟ๐‘กโˆ’1๎€œ๎‚ต๎‚ธ๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2(๎€œ๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œโˆ’๐‘ก0๐น(๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ๎‚น๎‚ถ.(3.9)
By subtracting relation (3.9) from (3.8),๐‘ข๐‘›+1(๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘ก)=๐‘ข๐‘›(๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐ฟ๐‘กโˆ’1๎‚ต๐‘ข๐‘›(๎€œ๐‘ฅ,๐‘ก)โˆ’๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘ก0๎€บ๐ท2๎€ท๐‘ข๐‘›(๎€ธ๐‘ฅ,๐œ)โˆ’๐ท2(๎€ปโˆ’๎€œ๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ๐‘ก0๎€บ๐น๎€ท๐‘ข๐‘›(๎€ธ๎€ป๎‚ถ,๐‘ฅ,๐œ)โˆ’๐น(๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ(3.10) if we set, ๐‘’๐‘›+1(๐‘Ÿ,๐‘ก)=๐‘ข๐‘›+1(๐‘Ÿ,๐‘ก)โˆ’๐‘ข๐‘›(๐‘Ÿ,๐‘ก), ๐‘’๐‘›(๐‘Ÿ,๐‘ก)=๐‘ข๐‘›(๐‘Ÿ,๐‘ก)โˆ’๐‘ข(๐‘Ÿ,๐‘ก), |๐‘’๐‘›(๐‘Ÿ,๐‘กโˆ—)|=max๐‘ก|๐‘’๐‘›(๐‘Ÿ,๐‘ก)| then since ๐‘’๐‘› is a decreasing function with respect to ๐‘ก from the mean value theorem we can write, ๐‘’๐‘›+1(๐‘Ÿ,๐‘ก)=๐‘’๐‘›(๐‘Ÿ,๐‘ก)+๐ฟ๐‘กโˆ’1๎‚ตโˆ’๐‘’๐‘›(๎€œ๐‘Ÿ,๐‘ก)+๐‘ก0๎€บ๐ท2๎€ท๐‘ข๐‘›(๎€ธ๐‘ฅ,๐œ)โˆ’๐ท2(๎€ปโˆ’๎€œ๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ๐‘ก0๎€บ๐น๎€ท๐‘ข๐‘›(๎€ธ๎€ป๎‚ถ๐‘ฅ,๐œ)โˆ’๐น(๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œโ‰ค๐‘’๐‘›(๐‘Ÿ,๐‘ก)+๐ฟ๐‘กโˆ’1๎€บโˆ’๐‘’๐‘›(๐‘Ÿ,๐‘ก)+๐ฟ๐‘กโˆ’1||๐‘’๐‘›||๎€ท๐œˆ๎€ท๐ฟ(๐‘Ÿ,๐‘ก)1+๐‘‡๐ฟ2๎€ธ๎€ธ๎€ปโ‰ค๐‘’๐‘›(๐‘Ÿ,๐‘ก)โˆ’๐‘‡๐‘’๐‘›๎€ท๐‘€(๐‘Ÿ,๐œ‚)+๐‘‡๎…ž๐ฟ1+๐‘€๎…ž๐ฟ2๎€ธ๐ฟ๐‘กโˆ’1๐ฟ๐‘กโˆ’1||๐‘’๐‘›||||๐‘’(๐‘Ÿ,๐‘ก)โ‰ค(1โˆ’๐‘‡(1โˆ’๐›ผ))๐‘›๎€ท๐‘Ÿ,๐‘กโˆ—๎€ธ||,(3.11) where 0โ‰ค๐œ‚โ‰ค๐‘ก. Hence, ๐‘’๐‘›+1(๐‘Ÿ,๐‘ก)โ‰ค๐›ฝ|๐‘’๐‘›(๐‘Ÿ,๐‘กโˆ—)|.
Therefore,โ€–โ€–๐‘’๐‘›+1โ€–โ€–=maxโˆ€๐‘กโˆˆ๐ฝ||๐‘’๐‘›+1||โ‰ค๐›ฝmaxโˆ€๐‘กโˆˆ๐ฝ||๐‘’๐‘›||โ€–โ€–๐‘’โ‰ค๐›ฝ๐‘›โ€–โ€–.(3.12) Since 0<๐›ฝ<1, then โ€–๐‘’๐‘›โ€–โ†’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: ๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘š=0๐‘ข๐‘š(๐‘ฅ,๐‘ก)(3.13) where lim๐‘šโ†’โˆž๐‘ข๐‘š(๐‘ฅ,๐‘ก)=0.(3.14)
We can write,๐‘›๎“๐‘š=1๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1๎€ป(๐‘ฅ,๐‘ก)=๐‘ข1+๎€ท๐‘ข2โˆ’๐‘ข1๎€ธ๎€ท๐‘ข+โ‹ฏ+๐‘›โˆ’๐‘ข๐‘›โˆ’1๎€ธ=๐‘ข๐‘›(๐‘ฅ,๐‘ก).(3.15)
We have,lim๐‘›โ†’โˆž๐‘ข๐‘›(๐‘ฅ,๐‘ก)=0.(3.16)
So, using (3.16) and the definition of the nonlinear operator ๐ฟ, we haveโˆž๎“๐‘š=1๐ฟ๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1๎€ป๎ƒฌ(๐‘ฅ,๐‘ก)=๐ฟโˆž๎“๐‘š=1๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1๎€ป๎ƒญ(๐‘ฅ,๐‘ก)=0.(3.17) Therefore from (2.32), we can obtain that, โˆž๎“๐‘š=1๐ฟ๎€บ๐‘ข๐‘š(๐‘ฅ,๐‘ก)โˆ’๐œ’๐‘š๐‘ข๐‘šโˆ’1(๎€ป๐‘ฅ,๐‘ก)=โ„Ž๐ป(๐‘ฅ,๐‘ก)โˆž๎“๐‘š=1โ„œ๐‘šโˆ’1๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๐‘ฅ,๐‘ก)=0.(3.18)
Since โ„Žโ‰ 0 and ๐ป(๐‘ฅ,๐‘ก)โ‰ 0, we haveโˆž๎“๐‘š=1โ„œ๐‘šโˆ’1๎€ท๐‘ข๐‘šโˆ’1(๎€ธ๐‘ฅ,๐‘ก)=0.(3.19)
By substituting โ„œ๐‘šโˆ’1(๐‘ข๐‘šโˆ’1(๐‘ฅ,๐‘ก)) into the relation (3.19) and simplifying it, we haveโˆž๎“๐‘š=1โ„œ๐‘šโˆ’1๎€ท๐‘ข๐‘šโˆ’1(๎€ธ=๐‘ฅ,๐‘ก)โˆž๎“๐‘š=1๎‚ธ๐‘ข๐‘šโˆ’1๎€œ(๐‘ฅ,๐‘ก)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€œ(๐‘ฅ,๐œ)๐‘‘๐œโˆ’๐‘ก0๐น๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€ท(๐‘ฅ,๐œ)๐‘‘๐œโˆ’1โˆ’๐œ’๐‘š๎€ธ๎‚น๎€œ๐‘“(๐‘ฅ)=๐‘ข(๐‘ฅ,๐‘ก)โˆ’๐‘“(๐‘ฅ)โˆ’๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1๎€ธโˆ’๎€œ(๐‘ฅ,๐œ)๐‘‘๐œ๐‘ก0๎€บ๐น๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€ป.(๐‘ฅ,๐œ)๐‘‘๐œ(3.20)
From (3.19) and (3.20), we have๎€œ๐‘ข(๐‘ฅ,๐‘ก)=๐บ(๐‘ฅ,๐‘ก)+๐‘ก0(๐‘กโˆ’๐œ)๐ท2(๎€œ๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œโˆ’๐‘ก0(๐‘กโˆ’๐œ)๐น(๐‘ข(๐‘ฅ,๐œ))๐‘‘๐œ,(3.21) therefore, ๐‘ข(๐‘ฅ,๐‘ก) must be the exact solution of (1.1).

Theorem 3.5. If |๐‘ข๐‘š(๐‘ฅ,๐‘ก)|โ‰ค1, then the series solution (2.46) of problem (1.1) converges to the exact solution.

Proof. We can write the solution ๐‘ข(๐‘ฅ,๐‘ก) as follows: ๐‘ข(๐‘ฅ,๐‘ก)=โˆž๎“๐‘š=0๐‘ข๐‘š(=๐‘ฅ,๐‘ก)โˆž๎“๐‘š=0๎‚ป๎€œ๐‘ก0๐ท2๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€œ(๐‘ฅ,๐œ)๐‘‘๐œ+๐‘ก0๐น๎€ท๐‘ข๐‘šโˆ’1๎€ธ+๎€ท(๐‘ฅ,๐œ)๐‘‘๐œ1โˆ’๐œ’๐‘š๎€ธ๎€œ๐‘“(๐‘ฅ)+๐‘ก0(๐‘กโˆ’๐œ)๐‘”1๎€ท๐‘ข๐‘šโˆ’1๎€ธ๎€œ(๐‘ฅ)๐‘‘๐œ+๐‘ก0๎€ท๐‘”(๐‘กโˆ’๐œ)2๎€ธ๎‚ผ.(๐‘ฅ)๐‘‘๐œ(3.22)
If |๐‘ข๐‘š(๐‘ฅ,๐‘ก)|<1, therefore, โˆ‘๐‘ข(๐‘ฅ,๐‘ก)=โˆž๐‘š=0๐‘ข๐‘š(๐‘ฅ,๐‘ก) 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 algorithm ๐œ€ is a given positive value.

Algorithm 4.1. One has the following.Step 1. Set ๐‘›โ†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 |๐‘ข๐‘›+1โˆ’๐‘ข๐‘›|<๐œ€ then go to Step 4, else ๐‘›โ†๐‘›+1 and go to Step 2.Step 4. Print โˆ‘๐‘ข(๐‘ฅ,๐‘ก)=๐‘›๐‘–=0๐‘ข๐‘–(๐‘ฅ,๐‘ก) as the approximate of the exact solution.

Example 4.2 (see [3]). Consider the Fisher equation with ๐‘ =3. ๐‘ข๐‘ก=๐‘ข๐‘ฅ๐‘ฅ๎€ท+๐‘ข1โˆ’๐‘ข3๎€ธ,(4.1) subject to initial conditions โŽ›โŽœโŽœโŽœโŽ1๐‘ข(๐‘ฅ,0)=๎‚€1+๐‘’โˆš(3/10)๐‘ฅ๎‚1/3โŽžโŽŸโŽŸโŽŸโŽ 2,(4.2) with the exact solution is โˆš{(1/2)tanh[โˆ’(3/2โˆš10)(๐‘ฅโˆ’(7/10)๐‘ก)]+(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.

tab1
Table 1: Numerical results of Example 4.2.

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