**Some Applications of the HAM**

**(1) Nonlinear ODEs**

Many nonlinear
boundary-value/eigen-value problems governed by

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

in
a finite or an infinite interval can be solved by means of the Mathematica
package **BVPh 1.0**. 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).

**(3) Interaction of
Nonlinear Water Wave and Non-uniform Current**

In **Chapter
15**, we illustrate the validity of the homotopy
analysis method
(HAM) for a complicated **nonlinear** **PDE** describing the nonlinear
interaction of a periodic traveling wave on a non-uniform current with
exponential distribution of vorticity.
In the frame of the HAM, the original highly **nonlinear** **PDE **with **variable coefficient** is transferred
into an infinite number of much simpler linear PDEs, which are rather easy to
solve. Physically, it is found
that StokesÕ criterion of wave breaking is still correct for traveling waves on
non-uniform currents. It verifies
that the HAM can be used to solve some complicated nonlinear PDEs so as to
deepen and enrich our physical understanding about some interesting nonlinear
phenomena.

For details, please refer to **Chapter 15** of LiaoÕs book:
**Homotopy Analysis Method in Nonlinear Differential Equations**,
Higher Education Press & Springer (2012).

**(4) Resonance Criterion
of Arbitrary Number of Nonlinear Water Waves **

In **Chapter
16**, we verify the validity of the homotopy
analysis method
(HAM) for a rather complicated **nonlinear
PDE** describing the nonlinear interaction of arbitrary number of traveling
water waves. In the frame of the
HAM, the wave resonance criterion for arbitrary number of waves is gained, for
the first time, which logically contains the famous PhillipsÕ criterion for
four small amplitude waves. Besides,
it is found for the first time that, when the wave-resonance criterion is satisfied
and the wave system is fully developed, there exist multiple steady-state resonant
waves, whose amplitude might be much smaller than primary waves so that a
resonant wave may contain much small percentage of the total wave energy. This example illustrates that the HAM
can be used as a tool to deepen and enrich our understandings about some rather
complicated nonlinear phenomena.

For details, please refer to **Chapter 16** of LiaoÕs book:
**Homotopy Analysis Method in Nonlinear Differential Equations**,
Higher Education Press & Springer (2012).