According to a recent analysis ofEssential Science Indicators from
Thomson Scientific,
Professor Ji-Huan He has been named aRising Starin the field of Computer
Science. His citation record in this field includes 21
papers cited a total of 306 times between January 1,
1997 and December 31, 2007. He also has 25 papers cited
a total of 881 times in Engineering, and 16 papers
cited a total of 87 times in Materials Science. His
citation record in the Web of
Science^{®} includes 137 papers cited a
total of 3,193 times to date.

Professor He is affiliated with Donghua University in Shanghai, China.
He is also the Founder and Editor-in-Chief of the International
Journal of Nonlinear Sciences and Numerical Simulation.

In the interview below, Professor He talks
with ScienceWatch.com about his highly cited
work.

Please tell us a little about your research and
educational background.

I studied Construction Engineering in the middle 1980s in Xi’an
University of Architecture & Technology, China, and received my
master's degree of mechanical engineering in 1990 from Shanghai University,
China—my thesis was Reliability Analysis of Pneumatical
Cylinders. Subsequently I worked as an engineer in two manufactories
for about three years. In late 1993 I became a Ph.D. candidate at Shanghai
University where I studied aerodynamics and calculus of variations. In my
Ph.D. thesis (defended in 1997) I proposed a new method called the
semi-inverse method to search for variational principles in fluids.

At that time, the finite element method became popular in China, and there
was not a universal approach to establishment of a variational formulation
directly from the governing equations and boundary/initial conditions. The
Lagrange multiplier method is the most-used method, but the method becomes
invalid for some special cases (e.g. the multiplier is vanishing). The
semi-inverse method was originally suggested to eliminate the demerit of
the Lagrange multiplier method, and it has become a useful mathematical
tool to the establishment of a variational formulation for a real-life
problem.

"The method
deforms a complex problem under
study to a series of linear
equations easy to be
solved."

After graduation, I focused myself on variational theory for smart material
and fluid mechanics, and then I turned my interest to analytical methods
for nonlinear equations, and suggested some new approximate analytical
methods, e.g., the variational iteration method, the homotopy perturbation
method, and the parameter-expansion method, which are now widely used to
solve various nonlinear equations.

In 2002, I moved to Donghua University doing research work on
nanotechnology. Our group devised some new devices for producing
nanofibers, such as vibration-electrospinning and magneto-electrospinning.
Just few months ago, we mimicked the possible mechanism of spider-spinning,
and suggested a new method called bubble-electrospinning for producing
nanofibers with high-throughput.

I am also interested in biology and high-energy physics, and have published
some papers on allometric scaling and E-infinity theory.

The majority of your highly cited papers deal with the
homotopy perturbation method (HPM)—what exactly is HPM?

The homotopy or the topology? It is elusive! An engineer or a
non-mathematics student might think so. However, the homotopy perturbation
method itself is simple enough to be mastered by a college student. The
method deforms a complex problem under study to a series of linear
equations easy to be solved.

Imagine a nonlinear beam supported on both ends with uniform loading; the
deflection is parabolic. We begin with a parabolic trial-function with some
unknown parameters, and construct a linear differential equation whose
solution is the chosen trial-function, then construct such a homotopy that
when the homotopy parameter p=0, it becomes the above constructed linear
equation; and when p=1, it turns out to be the original nonlinear equation.
The changing process of p from zero to unity is just that of the
trial-function (initial solution) to the exact solution. To approximately
solve the problem, the solution is expanded into a series of p, just like
that of the classical perturbation method. Generally one iteration is
enough, the unknown parameters can be determined optimally in view of
physical understanding after the iteration procedure is finished.

Hereby I will illustrate the general solution procedure of the method.
Consider a nonlinear equation in the form Lu+Nu=0, (1)
where L and N are linear operator and nonlinear operator respectively. In
order to use the homotopy perturbation, a suitable construction of a
homotopy equation is of vital importance. Generally a homotopy can be
constructed in the form L_{1}u+p(Lu+Nu-L_{1}u)=0
(2)
where L_{1} can be a linear operator or a simple nonlinear operator
, and the solution of L_{1}u=0 with possible some unknown
parameters can best describe the original nonlinear system. For example for
a nonlinear oscillator we can choose L_{1}u=ü+?^{2}u,
where ? is the frequency of the nonlinear oscillator to be further
determined.

I hope this explanation is enough for a beginner to use the method to solve
practical problems.

What are the applications for HPM?

The applications of the homotopy perturbation method mainly cover in
nonlinear differential equations, nonlinear integral equations, nonlinear
differential-integral equations, difference-differential equations, and
fractional differential equations.

Would you give our readers some examples of these
applications?

I will use a simple example to illustrate the solution procedure:

1) Mathematical model We consider a simple
mathematical model for a reaction-diffusion process in the form (see
Lu-Feng Mo, "Variational approach to reaction-diffusion process,"
Physics Letters A 368[3-4]: 263-5, August
2007) u"+u^{2}=0, u(0)=u(1)=0 (3)

2) Qualitative sketch/trial function solution This is
a boundary value problem, so we choose such an initial guess u_{0}(t)=at(1-t)
(4)
where a is an unknown constant. The trial-function, Eq.(4), satisfies the
boundary conditions.

3) Construction of a homotopy According to the
initial guess, a homotopy should be constructed: u"+2a+p(u^{2}-2a)=0,
u(0)=u(1)=0 (5)
When p=0, the solution of Eq.(5) is Eq.(4); When p=1, it turns out to be
the original one.

4) Solution procedure similar to that of classical perturbation
method Using p as an expanding parameter as that in classic
perturbation method, we have u_{0}"+2a=0, u_{0}(0)=u(0),
u_{0}(1)=u(1) (6) u_{1}"+(u_{0})^{2}=0,
u_{1}(0)=u_{1}(1)=0 (7)
Generally we need only few items. Setting p=1, we obtain the first-order
approximate solution which reads u(t)=u_{0}(t)+u_{1}(t)=at(1-t)+at
^{2}-a^{2}(t^{6}/30-t^{5}/30+t^{4}/12)-(a-a
^{2}/60)/t (8)

5) Optimal identification of the unknown parameter in the trial
function There are many approaches to identification of the
unknown parameters in the obtained solution. For periodic solution we can
identify the unknown parameter in view of no secular terms in the final
solution; for exponential solution, we can eliminate the terms
t^{n}exp(at) to identify the unknown parameters. We suggest hereby
the method of weighted residuals, especially the least squares
method:
int(R^{2},t,0,1) is a minimum where R is the residual
R(u(t))=Lu+Nu=u"+u^{2}, int(R^{2},t,0,1) denotes the
definitive integral of R^{2} with respect to t from 0 to 1

What are the advantages and disadvantages of HPM
compared with other available methods?

The obvious advantage of the method is that it can be applied to various
nonlinear problems. The main disadvantage is that we should suitably choose
an initial guess, or infinite iterations are required.

Where do you see this field going in five to ten
years?

Combination of numerical methods, e.g. the finite element method, with the
present method can solve some more complex problems, such as inverse
problems, computer vision, and image processing. I should emphasize that
the homotopy perturbation method might be a powerful tool for inverse shape
design. The boundary can be assumed in the form:
B(x,y,z)=(1-p)f(x,y,z)+pg(x,y,z), where f is an initial guess, and g is the
searched boundary.

What should the "take-away lesson" about your work be
for the general public—what would you like them to know about
your work?

Using the homotopy perturbation method with physical understanding, this is
my message for the general public: the method is under development, many
modified versions were appeared in literature, most are reasonable.

For a beginner, I suggest the following publications:

He JH, "New interpretation of homotopy perturbation method,"
International Journal Of Modern Physics B 20(18): 2561-68, 2006.

He JH, "Some asymptotic methods for strongly nonlinear equations,"
International Journal Of Modern Physics B 20(10): 1141-99,
2006.

Professor JH He
Donghua University
Shanghai, People's Republic of China

Professor
JH He's most-cited paper with 146
cites to date:

He JH, “Variational iteration method—a kind on
non-linear analytical technique: some examples,”
Int. J. Non-Linear Mech. 34(4): 699-708, July
1999. Source:
Essential Science Indicators from
Thomson
Scientific.