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.

Fisher proposed equation

The paper is organized as follows. In Section

To obtain the approximation solution of (

We set

In (

The terms

We set

Now we decompose the unknown function

The Adomian decomposition method is applied to the following general nonlinear equation:

The standard decomposition technique represents the solution of

The modified decomposition method was introduced by Wazwaz in [

To obtain the approximation solution of (

The operators

From [

In the VIM [

The VIM has been shown to solve effectively, easily and accurately a large class of nonlinear problems with approximations converge rapidly to accurate solutions.

To obtain the approximation solution of (

To find the optimal

From (

Therefore, the Lagrange multipliers can be identified as

Relation (

Consider

It should be emphasized that we have great freedom to choose the initial guess

Enforcing the homotopy (

we have the so-called zero-order deformation equation

When

Thus, according to (

Due to Taylor's theorem,

Let the initial guess

From (

By differentiating (

Therefore,

Note that the high-order deformation (

To obtain the approximation solution of (

Substituting (

We take an initial guess

Therefore, the solution

To explain MHPM, we consider (

The MHPM uses the homotopy parameter

Let

Let

From which we get

The series solution

Denote as

From [

So,

Let

From the triangle inquality we have

Since

But

The series solution

One has the following:

By subtracting relation (

Therefore,

If the series solution (

We assume:

We can write,

We have,

So, using (

Since

By substituting

From (

If

We can write the solution

If

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

One has the following.

Set

Calculate the recursive relation (

If

Print

Consider the Fisher equation with

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.

Numerical results of Example

Error | Error | Error | Error | |

(MADM, | (VIM, | (HAM, | (MHPM, | |

0.0437778 | ||||

0.068559 | 0.0670032 | |||

0.0558752 | ||||

0.9 | 0.0745342 | 0.0746331 | 0.0566234 | 0.07458943 |

1.0 | 0.0766331 | 0.0775012 | 0.0599735 | 0.0775367 |