FAST BREAKING PAPERS

Why do you think your paper is highly cited?

I think there may be several reasons why this particular paper is highly cited. I should categorize it in two sections:

1. The method: First, my paper uses a relatively new method proposed by a famous Chinese mathematician—Dr. Ji-Huan He of Shanghai University—in order to solve a complex nonlinear problem, which was very difficult to solve by the use of traditional methods, such as Adomian and classical perturbation.

Secondly, the solution procedure is of remarkably simplicity and even the first-order approximate solutions are always of extreme accuracy. Thirdly, the obtained solutions are valid for the entire solution domain.

Fourthly, Dr. He’s homotopy perturbation method (HPM) is a universal mathematical tool for solving nonlinear problems. I anticipate that He’s HPM, without any prerequisites of the small parameter assumption and linearization, will become as popular as Newton’s iteration method in future, and will be widely used by engineers and scientists as a mathematical tool for revealing hidden physical meanings in various nonlinear equations.

2. The application: this application, i.e., that of the reaction-diffusion equation, is a kind of universal problem. This equation has been used when a fluid flows into soil—a general and complex problem in the area of civil engineering. In mechanical engineering, when a fluid flows in porous media, a complex equation is also involved. In chemical engineering, when a colored material mixes with another material, the problem recurs.

This equation is hard to solve. A numerical algorithm is almost the only way to tackle this problem. I have successfully shown an analytical method of treatment for the very first time.

Does it describe a new discovery, methodology, or synthesis of knowledge?

Yes, the paper does describe several new discoveries of new applications of Dr. He’s HPM. Previously, many methods were used to study the same problem, but solution procedures are always complex and burdensome, and the results often deteriorate quickly as the degree of nonlinearity increases. Now, things have begun to change, and He’s HPM can completely surpass the shortcomings of many famously known methods, such as the Adomian method.

Would you summarize the significance of your paper in layman’s terms?

He’s perturbation method itself is mathematically beautiful and extremely accessible to non-mathematicians. The use of the method requires no special knowledge of elusive topology. The method deforms a complex problem under study to a simple, routine problem. Generally, one iteration is enough, making the method a most attractive one.

How did you become involved in this research, and were there any problems along the way?

In his review article (He JH, "Some asymptotic methods for strongly nonlinear equations," Int. J. Mod. Phys. B, 20 [10]: 1141-1199, 2006), Ji-Huan He gave a very lucid as well as elementary discussion of the asymptotic techniques for various nonlinear equations, including the variational iteration method and the homotopy perturbation method.

Where do you see your research leading in the future?

Wherever a nonlinear equation is found, Dr. He’s HPM will be the primary tool of discovery.

Davood Domiri Ganji, Ph.D.
Department of Mechanical Engineering
Nushirvani Engineering Complex
Mazandaran University
Babol, Mazandaran, Iran