Most interesting phenomena in physics, social sciences, engineering, and other
disciplines are highly non-linear. This limits the ability to analytically
investigate such systems. Simulations of the dynamical processes are then the tool of
choice to explore the system. However, it is sometimes very important to have a
basic understanding in terms of approximative solutions. Non-linear differential
equations describing the dynamics are known to be harder to solve then
One often has to resort to asymptotic techniques or classical perturbation theory to obtain analytical approximations.
Classical perturbation theory strongly depends on small/large physical
parameters. Therefore, such methods are only valid for weakly non-linear systems.
Homotopy Analysis Method is a quite new approach to explore highly non-linear systems. The method composes the non-linear system by linear parts and approximates the ‘real’ solution by an iterative process. The convergence speed is governed by a tuning
parameter q. The approximative solution then can be found as a linear combination of
base functions. The main advantage of HAM are:
- Independence of small/large physical parameters
- Flexibility on the choice of the base functions
Moreover, if for a perturbative method the convergence is guaranteed in a small interval 0<=e<=1, then the HAM method allows the convergence in the whole interval 0 <= e < N, with N >> e and arbitrary.
To give an impression of the method we apply it to a rather simple non-linear problem, namely the susceptible infected disease model (SI). Albeit there exists an exact solution for that problem it is illustrative to see the HAM technique at work. As a base function we have chosen an exponential function respecting the initial condition of the problem (stated in the code).
The code can be downloaded here. A comprehensive treatment of the HAM technique can be found in Shijun Liao’s book.