**Homotopy Analysis Method**

**Basic
Ideas & Brief History**** ****Publications**** ****Examples** **Mathematica Package BVPh 1.0** **Mathematica Package APOh** **Maple
Package NOPH 1.0**

The
homotopy analysis method (HAM) is an analytic approximation method for highly
nonlinear equations in science, finance and engineering. It was first proposed by Dr. Shijun
LIAO in 1992
in his PhD dissertation, and modified and developed by Dr. Liao with his team
and researchers in many other countries.

**Motivations**

Perturbation
techniques are widely applied to gain analytic approximations of nonlinear
equations. However, perturbation
methods are essentially based on small physical parameters (called perturbation
quantity), but unfortunately many nonlinear problems have **no** such kind of small physical parameters at all. In addition, **neither** perturbation techniques **nor** the traditional non-perturbation techniques (such as Lyapunov
artificial small-parameter method, Adomian decomposition method,
delta-expansion method and so on) can provide a way to guarantee the
convergence of approximation series.
Therefore, both perturbation techniques and the traditional non-perturbation
methods mentioned above are in essence valid only for weakly nonlinear
problems.

**Advantages of the homotopy analysis method**

Based on
a generalized concept of the homotopy in topology, the HAM has the following
advantages:

·
The HAM
is always valid **no matter** **whether** there exist small physical
parameters **or not**;

·
The HAM
provides a convenient way to **guarantee**
the convergence of approximation series;

·
The HAM
provides **great freedom** to choose the
equation type of linear sub-problems and the base functions of solutions.

As a result, the HAM
overcomes the restrictions of all other analytic approximation methods
mentioned above, and is valid for highly nonlinear problems.

**Some new concepts and theorems**

Dr. Liao
proposed some new concepts in the frame of the HAM, as mentioned below.

**1.
****Convergence Control**

In the
frame of the HAM, the so-called **convergence-control
parameter** is introduced **for the
first time **by Dr. Liao in 1997.
This auxiliary parameter has no physical meanings, but provides us a
convenient way to **guarantee **the
convergence of approximation series.
In 1999, Dr. Liao further generalized the HAM by introducing
more such kind of auxiliary parameters to enlarge the ability of guaranteeing
the convergence of approximation series.

**2.
****Rule of Solution Expression**

Based on
a generalized concept of homotopy proposed by Dr. Liao in 1997, the HAM transfers a
nonlinear problem into an infinite number of linear sub-problems with great
freedom to choose the corresponding equation-type and solution expressions. Unlike perturbation techniques, the
starting-point of the HAM is to choose a proper, complete set of base functions
to express solutions, according to the physical background or the given
boundary/initial conditions or asymptotic properties of considered problems. The so-called zeroth-order deformation
equation (which is a kind of homotopy) should be constructed in such a way that
every equation of high-order approximations has a unique solution that obeys
the given solution-expression. This
is so important that it becomes a rule in the frame of the HAM, i.e. the rule
of solution expression.

The Rule
of Solution Expression simplifies the use of the HAM, since it provides a guide
to choose the equation type and base functions of linear sub-problems. Besides, using the Rule of Solution
Expression, many unknown physical parameters (such as the period of a nonlinear
oscillator) can be easily obtained.
In addition, using the Rule of Solution Expression, some equations of
high-order approximations can be solved in a much easier way.

**3.
****Homotopy Transform**

In the
frame of the HAM, Dr. Liao deduced a kind of new transform, called the
homotopy transform, and further proved that the famous Euler transform is only
a special case of the homotopy transform.
This provides us a mathematical cornerstone for the reasonableness of
the HAM and for its general validity.