Introduction to Mathematical Physics/Some mathematical problems and their solution/Nonlinear evolution problems, perturbative methods
Problem statement edit
Perturbative methods allow to solve nonlinear evolution problems. They are used in hydrodynamics, plasma physics for solving nonlinear fluid models (see for instance ([ph:plasm:Chen84]). Problems of nonlinear ordinary differential equations can also be solved by perturbative methods (see for instance ([ma:equad:Arnold83]) where averaging method is presented). Famous KAM theorem (Kolmogorov--Arnold--Moser) gives important results about the perturbation of hamiltonian systems. Perturbative methods are only one of the possible methods: geometrical methods, normal form methods ([ma:equad:Arnold83]) can give good results. Numerical technics will be introduced at next section.
Consider the following problem:
proeqp
Problem: Find such that:
- verifies boundary conditions on the border of .
- verifies initial conditions.
Various perturbative methods are presented now.
Regular perturbation edit
Solving method can be described as follows:
Algorithm:
- Differential equation is written as:
- The solution of the problem when is zero is known.
- General solution is sought as:
- Function is developed around using Taylor type formula:
- A hierarchy of linear equations to solve is obtained:
This method is simple but singular problem my arise for which solution is not valid uniformly in .
Example:
Non uniformity of regular perturbative expansions (see ([ma:equad:Bender87]). Consider Duffing equation:
Let us look for solution which can be written as:
The linear hierarchy obtained with the previous assumption is:
With initial conditions:
,
one gets:
and a particular solution for will be unbounded[1] , now solution is expected to be bounded. Indeed (see [ma:equad:Bender87]), multiplying Duffing equation by , one gets the following differential equation:
We have thus:
where is a constant. Thus is bounded if .
Remark:
In fact Duffing system is conservative.
Remark:
Origin of secular terms : A regular perturbative expansion of a periodical function whose period depends on a parameter gives rise automatically to secular terms (see ([ma:equad:Bender87]):
Born's iterative method edit
Algorithm:
- Differential equation is transformed into an integral equation:
- A sequence of functions converging to the solution is sought:
Starting from chosen solution , successive are evaluated using recurrence formula:
This method is more "global" than previous one \index{Born iterative method} and can thus suppress some divergencies. It is used in diffusion problems ([ph:mecaq:Cohen73],[ph:mecaq:Cohen88]). It has the drawback to allow less control on approximations.
Multiple scales method edit
Algorithm:
For examples see ([ma:equad:Nayfeh95]).
Poincaré-Lindstedt method edit
This method is closely related to previous one, but is specially dedicated to studying periodical solutions. Problem to solve should be:\index{Poincaré-Lindstedt}
Problem:
Find such that:
eqarespo
where is a periodic function of pulsation . Setting ,
one gets:
Resolution method is the following:
WKB method edit
mathsecWKB
WKB (Wentzel-Krammers-Brillouin) method is also a perturbation method. It will be presented at section secWKB in the proof of ikonal equation.
- ↑
Indeed solution of equation:
is