Davood Domiri Ganji talks
with ScienceWatch.com and answers a few questions
about this month's Fast Breaking Paper in the field of
Engineering.

Field: Engineering Article Title: Application of He's
homotopy-perturbation method to nonlinear coupled systems
of reaction-diffusion equations
Authors:
Ganji,
DD;Sadighi, A
Journal: INT J NONLINEAR SCI NUMER SIM
Volume: 7
Issue: 4
Page: 411-418
Year: 2006
* Mazandaran Univ, Dept Engn Mech, POB 484, Babol Sar,
Iran.
* Mazandaran Univ, Dept Engn Mech, Babol Sar, Iran.
(addresses may have been truncated; see full
article)

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.

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

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