**Education**** ****Work
Experience**** ****Publications**** ****Research
Interests** **Awards HAM BVPh ** **APOh
****Free-downloaded
codes**** **

**Research Interests of ****Dr. Liao**** in Applied Mathematics**

**(1) HAM-Based
Mathematica package BVPh**

Based on the homotopy analysis
method (HAM), Dr.
Liao developed a Mathematica package BVPh
(version 1.0) for highly nonlinear boundary-value/eigen-value problems governed
by

*F*[*u*(*x*)
, *x*] = 0

in
a finite or an infinite interval.
Here, the governing equation may contain singularity and have multiple
solutions, and the boundary condition may be satisfied at multiple points.

Many examples
are given in **Chapters 8 – 12**
of LiaoÕs book: **Homotopy Analysis Method in Nonlinear
Differential Equations**, Higher
Education Press & Springer (2012)

**(2) American Put
Option**

American
put option is governed by a **partial
differential equation (PDE)** with an unknown moving boundary (i.e. the
optimal exercise boundary). The homotopy
analysis method (HAM) is successfully applied to solve this famous problem in
finance. Unlike asymptotic and/or
perturbation formulas that are often valid only a couple of days or weeks prior
to expiry, the optimal exercise boundary given by the HAM may be valid a couple
of dozen years, or even a half century!
This illustrates the great potential and general validity of the HAM for
nonlinear PDE. A practical **Mathematica Package APOh** with a simple userÕs guide is provided for businessmen to
gain accurate enough optimal exercise price of American put option at large
expiration-time by a laptop only in a few seconds, which is free available at http://numericaltank.sjtu.edu.cn/APO.htm.

For details, please refer to **Chapter 13** of LiaoÕs book:
**Homotopy Analysis Method in Nonlinear Differential Equations**,
Higher Education Press & Springer (2012).