Perturbation theory: Encyclopedia II - Perturbation theory - Simple example

Perturbation theory - Simple example

Consider the following equation for the unknown variable x:

For the initial problem with ε = 0, the solution is x₀ = 1. For small ε the lowest order approximation may be found by inserting the ansatz

into the equation and demanding the equation to be fulfilled up to terms that involve powers of ε higher than the first. This yields x₁ = 1. In the same way, the higher orders may be found. However, even in this simple example it may be observed that for (arbitrarily) small ε > 0 there are four other solutions to the equation (with very large magnitude). The reason we don't find these solutions in the above perturbation method is because these solutions diverge when while the ansatz assumes regular behavior in this limit.

The four additional solutions can be found using the methods of singular perturbation theory. In this case this works as follows. Since the four solutions diverge at ε = 0, it makes sense to rescale x. We put

such that in terms of y the solutions stay finite. This means that we need to choose the exponent ν to match the rate at which the solutions diverge. In terms of y the equation reads:

The 'right' value for ν is obtained when the exponent of ε in the prefactor of the term proportional to y is equal to the exponent of ε in the prefactor of the term proportional to y⁵, i.e. when ν = 1 / 4. This is called 'significant degeneration'. If we choose ν larger then the four solutions will collapse to zero in terms of y and they will become degenerate with the solution we found above. If we choose ν smaller then the four solutions will still diverge to infinity.

Putting ν = 1 / 4 in the above equation yields:

This equation can be solved using ordinary perturbation theory in the same way as regular expansion for x was obtained. Since the expansion parameter is now ε^{1 / 4} we put:

There are 5 solutions for y₀: 0, 1, -1, i and -i. We must disregard the solution y = 0. The case y = 0 corresponds to the original regular solution which appears to be at zero for ε = 0, because in the limit we are rescaling by an infinite amount. The next term is y₁ = − 1 / 4. In terms of x the four solutions are thus given as:

Both regular and singular perturbation theory are frequently used in physics and engineering. Regular perturbation theory may only be used to find those solutions of a problem that evolve smoothly out of the initial solution when changing the parameter (that are "adiabatically connected" to the initial solution). A well known example from physics where regular perturbation theory fails is in fluid dynamics when one treats the viscosity as a small parameter. Close to a boundary, the fluid velocity goes to zero, even for very small viscosity (the no-slip condition). For zero viscosity, it is not possible to impose this boundary condition and a regular perturbative expansion amounts to an expansion about an unrealistic physical solution. Singular perturbation theory can, however, be applied here and this amounts to 'zooming in' at the boundaries (boundary layer theory).

Perturbation theory can fail when the system can go to a different "phase" of matter, with a qualitatively different behaviour that cannot be understood by perturbation theory (e.g., a solid crystal melting into a liquid). In some cases this failure manifests itself by divergent behavior of the perturbation series. Such divergent series can sometimes be resummed using techniques such as Borel resummation.

Perturbation techniques can be also used to find approximate solutions to non-linear differential equations. Examples of techniques used to find approximate solutions to these types of problems are the Lindstead-Poincaré technique, the method of harmonic balancing, and the method of multiple time scales.

Adapted from the Wikipedia article "Simple example", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki