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Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 795838, 5 pages
Homotopy Perturbation Method for Solving Reaction-Diffusion Equations
School of Information Engineering, Zhejiang Forestry College, Lin####^~^~^~^~^~^####x27;an 311300, Zhejiang, China
Received 16 November 2007; Revised 11 February 2008; Accepted 27 February 2008
Academic Editor: Paulo Gon####^~^~^~^~^~^####xE7;alves
Copyright © 2008 Yu-Xi Wang et al. 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.
The homotopy perturbation method is applied to solve reaction-diffusion equations. In this method, the trial function (initial solution) is chosen with some unknown parameters, which are identified using the method of weighted residuals. Some examples are given. The obtained results are compared with the exact solutions, revealing that this method is very efficient and the obtained solutions are of high accuracy.
In this paper, we consider a reaction-diffusion process governed by the nonlinear ordinary differential equation : with boundary conditions where represents the steady-state temperature for the corresponding reaction-diffusion equation with the reaction term is the power of the reaction term (heat source), generally it follows is the length of the sample (heat conductor). The physical interpretation of (1.1) was given in .
Recently, various different analytical methods were applied to nonlinear equations arising in engineering applications, such as the homotopy perturbation method [2####^~^~^~^~^~^####x2013;10], and exp-function method [11, 12], a complete review is available in . This problem was studied by Lesnic using Adomian method , and by Mo  using variational method. In this paper, the homotopy perturbation method [2, 3, 13] is applied to the discussed problem, and the obtained results show that the method is very effective and simple.
2. Homotopy Perturbation Method
In order to use the homotopy perturbation, we construct a homotopy in the form [2, 3, 13] with initial approximation where is an unkown constant to be further determined. It is obvious that (2.2) satisfies the boundary conditions.
We rewrite (2.1) in the form of We suppose the solution of (2.3) has the form Substituting (2.4) into (1.1) and equating the terms with the identical powers of , we can solve sequentially with ease. Setting , we obtain the approximate solution of (1.1) in the form of To illustrate its solution procedure, we consider some special cases.
Case 1 (). Under such case, we can easily obtain sequentially We, therefore, obtain the approximate solution in the form of In order to identify the unknown constant in (2.7), we apply the method of weighted residuals. Subsituting (2.7) into (1.1) results in the following residual: It is obvious that and . We locate at , and set , yielding the result
Case 2 (). The solution procedure is the same as that for Case 1. We can easily obtain the following linear equations: We obtain the following second-order approximate solution: Similarly, we locate at , and set to identify the unknown constant, which reads .
Case 3 (). By the same manuplation as illustrated in above cases, we obtain Using the method of weighted residuals, we set , resulting in .
Figure 1 shows the remarkable accuracy of the obtained results.
The homotopy perturbation method deforms a complex problem under study to a simple problem routinely. If initial guess is suitably chosen, one iteration is enough, making the method a most attractive one. The method is of remarkable simplicity, while the obtained results are of utter accuracy on the whole solution domain. The method can be applied to various other nonlinear problems without any difficulty.
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