# Homotopy Perturbation Method for Solving Reaction-Diffusion Equations

Authors: Yu-Xi Wang; Hua-You Si; Lu-Feng Mo
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

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.


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## Body

## 1. Introduction

In this paper, we consider a reaction-diffusion process governed by the nonlinear ordinary
differential equation [1]: (1.1)y′′(x)+yn(x)=0,0<x<L, with boundary
conditions (1.2)y(0)=y(L)=0, where y(x) represents the steady-state
temperature for the corresponding reaction-diffusion
equation with the reaction term yn;n is the power of the reaction term
(heat source), generally it follows n>0,L is the length of the sample (heat conductor). The
physical interpretation of (1.1) was given in [1].Recently, various different analytical methods were
applied to nonlinear equations arising in engineering applications, such as the
homotopy perturbation method [2–10], and exp-function method [11, 12], a
complete review is available in [13]. This problem was studied by Lesnic using Adomian method [1], and
by Mo [14] 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] (2.1)(1−p)(y′′−y0′′)+p(y′′+yn)=0 with initial approximation (2.2)y0(x)=ax(1−x)=ax−ax2, where a 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(2.3)y′′+2a−p(2a−yn)=0. We suppose the solution of (2.3) has the form (2.4)y(x)=y0(x)+py1(x)+p2y2(x)+⋯. Substituting (2.4) into (1.1) and
equating the terms with the identical powers of p, we can solve y0,y1,y2,… sequentially with ease. Setting p=1, we obtain
the approximate solution of (1.1) in the form of(2.5)y(x)=y0(x)+y1(x)+y2(x)+⋯. To illustrate its solution procedure,
we consider some special cases.Case 1 (n=2).
Under such case, we can easily obtain
sequentially(2.6)y0′′=−2a,y1′′=2a−y02,y2′′=−2y0y1. We, therefore, obtain the approximate
solution in the form of (2.7)y(x)=ax(1−x)+ax2−a2(130x6−110x5+112x4)−(a−160a2)x. In order to identify the unknown
constant a in (2.7), we apply the method
of weighted residuals. Subsituting (2.7) into (1.1) results in the following
residual: (2.8)R(x,a)=y′′(x)+yn(x). It is obvious that R(0,a)=0 and R(1,a)=0. We locate at x=1/3, and set R(1/3,a)=0, yielding the result (2.9)a=45.4205.Case 2 (n=3).
The solution procedure is the same as
that for Case1. We can easily obtain the following linear equations: (2.10)a=14.2657,y1′′=2a−y03,y2′′=−3y02y1. We obtain the following second-order
approximate solution: (2.11)y(x)=ax(1−x)+ax2−a3(−156x8+114x7−110x6+120x5)−(a−1280a3)x. Similarly, we locate at x=1/3, and set R(1/3,a)=0 to identify the unknown constant, which reads a=14.2657.Case 3 (n=4).
By the same manuplation as
illustrated in above cases, we obtain(2.12)y0′′=−2a,y1′′=2a−y04,y2′′=−4y03y1,y(x)=ax(1−x)+ax2−a4(190x10−118x9+328x8−221x7+130x6)−(a−7.9410−4a4)x. Using the method of weighted residuals, we set R(1/3a)=0, resulting in a=9.6320.Figure1 shows the remarkable
accuracy of the obtained results.Comparison of approximate solutions with exact ones. Continued line:
approximate solution; discontinued line: exact solution.
(a)
n=2(b)
n=3(c)
n=4


## 3. Conclusion

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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