Homotopy Perturbation Method for Solving Reaction-Diffusion Equations

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.

1. Introduction

In this paper, we consider a reaction-diffusion process governed by the nonlinear ordinary differential equation [1]: 𝑦^′′(𝑥)+𝑦^𝑛(𝑥)=0,0<𝑥<𝐿, (1.1) with boundary conditions 𝑦(0)=𝑦(𝐿)=0, (1.2) where 𝑦(𝑥) represents the steady-state temperature for the corresponding reaction-diffusion equation with the reaction term 𝑦^𝑛;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] (1−𝑝)𝑦^′′−𝑦₀^′′+𝑝𝑦^′′+𝑦^𝑛=0 (2.1) with initial approximation 𝑦₀(𝑥)=𝑎𝑥(1−𝑥)=𝑎𝑥−𝑎𝑥², (2.2) 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 of𝑦^′′+2𝑎−𝑝2𝑎−𝑦^𝑛=0. (2.3) We suppose the solution of (2.3) has the form 𝑦(𝑥)=𝑦₀(𝑥)+𝑝𝑦₁(𝑥)+𝑝²𝑦₂(𝑥)+⋯. (2.4) 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 of𝑦(𝑥)=𝑦₀(𝑥)+𝑦₁(𝑥)+𝑦₂(𝑥)+⋯. (2.5) To illustrate its solution procedure, we consider some special cases.

Case 1 (𝑛=2). Under such case, we can easily obtain sequentially 𝑦₀^′′𝑦=−2𝑎,_1′′=2𝑎−𝑦²₀,𝑦₂^′′=−2𝑦₀𝑦₁. (2.6) We, therefore, obtain the approximate solution in the form of 𝑦(𝑥)=𝑎𝑥(1−𝑥)+𝑎𝑥²−𝑎²1𝑥30⁶−1𝑥10⁵+1𝑥12⁴−1𝑎−𝑎60²𝑥. (2.7) 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: 𝑅(𝑥,𝑎)=𝑦^′′(𝑥)+𝑦^𝑛(𝑥). (2.8) 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 𝑎=45.4205. (2.9)

Case 2 (𝑛=3). The solution procedure is the same as that for Case 1. We can easily obtain the following linear equations: 𝑦𝑎=14.2657,_1′′=2𝑎−𝑦³₀,𝑦₂^′′=−3𝑦²₀𝑦₁. (2.10) We obtain the following second-order approximate solution: 𝑦(𝑥)=𝑎𝑥(1−𝑥)+𝑎𝑥²−𝑎³−1𝑥56⁸+1𝑥14⁷−1𝑥10⁶+1𝑥20⁵−1𝑎−𝑎280³𝑥. (2.11) 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 𝑦₀^′′𝑦=−2𝑎,_1′′=2𝑎−𝑦⁴₀,𝑦₂^′′=−4𝑦³₀𝑦₁,𝑦(𝑥)=𝑎𝑥(1−𝑥)+𝑎𝑥²−𝑎⁴1𝑥90¹⁰−1𝑥18⁹+3𝑥28⁸−2𝑥21⁷+1𝑥30⁶−𝑎−7.9410⁻⁴𝑎⁴𝑥. (2.12) Using the method of weighted residuals, we set 𝑅(1/3𝑎)=0, resulting in 𝑎=9.6320.

Figure 1 shows the remarkable accuracy of the obtained results.

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Figure 1 

Comparison of approximate solutions with exact ones. Continued line: approximate solution; discontinued line: exact solution.

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.