## Professor JH He Interview

Rising Star in the field of Computer Science. Professor He's 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.*

**SW: 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.

**SW: 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**L1u+p(Lu+Nu-L1u)=0** (2)

where L1 can be a linear operator or a simple nonlinear operator , and the solution of L1u=0 with possible some unknown parameters can best describe the original nonlinear system. For example for a nonlinear oscillator we can choose L1u=ü+?2u, 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.

**SW: 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.

**SW: 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"+u2=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**u0(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(u2-2a)=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**u0"+2a=0, u0(0)=u(0), u0(1)=u(1)** (6)**u1"+(u0)2=0, u1(0)=u1(1)=0** (7)

Generally we need only few items. Setting p=1, we obtain the first-order approximate solution which reads**u(t)=u0(t)+u1(t)=at(1-t)+at 2-a2(t6/30-t5/30+t4/12)-(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 tnexp(at) to identify the unknown parameters. We suggest hereby the method of weighted residuals, especially the least squares method:

int(R2,t,0,1) is a minimum where R is the residual R(u(t))=Lu+Nu=u"+u2, int(R2,t,0,1) denotes the definitive integral of R2 with respect to t from 0 to 1

**SW: 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.

**SW: 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.

**SW: 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 Reuters.

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