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 as a model for the propagation of a mutant gene, with 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: with the initial conditions given by
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 and using the initial conditions, we obtain
We set
In (1.3), we assume is bounded for all in and
The terms and are Lipschitz continuous with , .
We set
Now we decompose the unknown function by a sum of components defined by the following decomposition series with identified as :
2. Iterative Methods
2.1. Preliminaries of the MADM
The Adomian decomposition method is applied to the following general nonlinear equation: where is the unknown function, is the highest-order derivative which is assumed to be easily invertible, is a linear differential operator of order less than represents the nonlinear terms, and is the source term. Applying the inverse operator to both sides of (2.1), and using the given conditions, we obtain where the function represents the terms arising from integrating the source term . The nonlinear operator is decomposed as where , are the Adomian polynomials determined formally as follows: Adomian polynomials were introduced in [12–15] as
2.1.1. Adomian Decomposition Method
The standard decomposition technique represents the solution of in (2.1) as the following series, where, the components are usually determined recursively by Substituting (2.5) into (2.7) leads to the determination of the components of . Having determined the components the solution 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 can be divided into two parts, namely and . Under this assumption we set Accordingly, a slight variation was proposed only on the components and . The suggestion was that only the part be assigned to the zeroth component , whereas the remaining part be combined with the other terms given in (2.7) to define . Consequently, the modified recursive relation 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:
The operators and are usually represented by an infinite series of the so-called Adomian polynomials as follows: where and are the Adomian polynomials were introduced in [12].
From [12], we can write another formula for the Adomian polynomials: where the partial sum is .
2.3. Preliminaries of the VIM
In the VIM [17–20], we consider the following nonlinear differential equation: where is a linear operator, is a nonlinear operator and is a known analytical function. In this case, a correction functional can be constructed as follows: where is a general Lagrange multiplier which can be identified optimally via variational theory. Here the function is a restricted variations which means . Therefore, we first determine the Lagrange multiplier that will be identified optimally via integration by parts. The successive approximation , of the solution will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function . The zeroth approximation may be selected any function that just satisfies at least the initial and boundary conditions. With determined, then several approximation , follow immediately. Consequently, the exact solution may be obtained by using
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: where,
To find the optimal , we proceed as
From (2.18), the stationary conditions can be obtained as follows:
Therefore, the Lagrange multipliers can be identified as and by substituting in (2.16), the following iteration formula is obtained.
Relation (2.20) will enable us to determine the components recursively for .
2.5. Preliminaries of the HAM
Consider where is a nonlinear operator, is unknown function and is an independent variable. let denote an initial guess of the exact solution , an auxiliary parameter, an auxiliary function, and an auxiliary nonlinear operator with the property when . Then using as an embedding parameter, we construct a homotopy as follows:
It should be emphasized that we have great freedom to choose the initial guess , the auxiliary nonlinear operator , the nonzero auxiliary parameter , and the auxiliary function .
Enforcing the homotopy (2.22) to be zero, that is,
we have the so-called zero-order deformation equation
When , the zero-order deformation (2.24) becomes and when , since and , the zero-order deformation (2.24) is equivalent to
Thus, according to (2.25) and (2.26), as the embedding parameter increases from 0 to 1, varies continuously from the initial approximation to the exact solution . Such a kind of continuous variation is called deformation in homotopy [21, 22].
Due to Taylor's theorem, can be expanded in a power series of as follows: where
Let the initial guess , the auxiliary nonlinear parameter , the nonzero auxiliary parameter and the auxiliary function be properly chosen so that the power series (2.27) of converges at , then, we have under these assumptions the solution series
From (2.27), we can write (2.24) as follows:
By differentiating (2.30) times with respect to , we obtain
Therefore, where,
Note that the high-order deformation (2.32) is governing the nonlinear operator , and the term can be expressed simply by (2.33) for any nonlinear operator .
2.6. Description of the HAM
To obtain the approximation solution of (1.1), according to HAM, let so
Substituting (2.36) into(2.32)
We take an initial guess , an auxiliary nonlinear operator , a nonzero auxiliary parameter , and auxiliary function . This is substituted into (2.37) to give the recurrence relation
Therefore, the solution becomes which is the method of successive approximations. If then the series solution (2.39) convergence uniformly.
2.7. Description of the MHPM
To explain MHPM, we consider (1.1) as where and . We can define homotopy by where is an unknown real number and Typically we may choose a convex homotopy by where is called the accelerating parameters, and for we define , which is the standard HPM. The convex homotopy (2.44) continuously trace an implicity defined curve from a starting point to a solution function . The embedding parameter monotonically increase from o to 1 as trivial problem is continuously deformed to original problem . [23, 24]
The MHPM uses the homotopy parameter as an expanding parameter to obtain when (2.44) corresponds to the original one, (2.45) becomes the approximate solution of (1.1), that is, where,
3. Existence and Convergency of Iterative Methods
Theorem 3.1. Let , then Fisher equation (1.1), has a unique solution.
Proof. Let and be two different solutions of (1.3) then
From which we get . Since . then . Implies and completes the proof.
Theorem 3.2. The series solution of problem (1.1) using MADM convergence when , .
Proof. Denote as the Banach space of all continuous functions on with the norm , for all in . Define the sequence of partial sums , and let and be arbitrary partial sums with . We are going to prove that is a Cauchy sequence in this Banach space:
From [12], we have
So,
Let , then
From the triangle inquality we have
Since , we have , then
But , so, as , then . We conclude that is a Cauchy sequence in , therefore the series is convergence and the proof is complete.
Theorem 3.3. The series solution of problem (1.1) using VIM converges when , .
Proof. One has the following:
By subtracting relation (3.9) from (3.8),
if we set, , , then since is a decreasing function with respect to from the mean value theorem we can write,
where . Hence, .
Therefore,
Since , then . 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:
where
We can write,
We have,
So, using (3.16) and the definition of the nonlinear operator , we have
Therefore from (2.32), we can obtain that,
Since and , we have
By substituting into the relation (3.19) and simplifying it, we have
From (3.19) and (3.20), we have
therefore, must be the exact solution of (1.1).
Theorem 3.5. If , then the series solution (2.46) of problem (1.1) converges to the exact solution.
Proof. We can write the solution as follows:
If , therefore, 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 is a given positive value.
Algorithm 4.1. One has the following.Step 1. Set .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 then go to Step 4, else and go to Step 2.Step 4. Print as the approximate of the exact solution.
Example 4.2 (see [3]). Consider the Fisher equation with . subject to initial conditions with the exact solution is .
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.