Talk:Singular perturbation

Definition
perhaps one should start with a difinition of a singularly perturbed problem?

can we agree on what that is? how about a problem involving a parameter ɛ so that as ɛ→0 the solution does not converge to the solution of the unperturbed problem (i.e. the problem where we set ɛ=0).

we can give a simple example ɛx^8 + x^2 -1 =0. There are 2 solutions that converge to the unperturbed problem, and 6 solutions that do not. This is a singularly perturbed problem.

Method of solution: find the possible scaling of x; for each scaling take the limit ɛ=0 and solve remaining problem. Convert back solution to original variables to reconstruct solution.

Ofcourse, this can be done also in ODE's and PDE's and possibly other fields...any nice examples anyone?

I think that the main point we should try to give is the importance of scaling for this method of solution.

--Yfarjoun 02:42, 15 February 2006 (UTC)

an example with ODEs
Here's a simple example with ODEs:

εy' ' + y' + y = 0, where y = y(x,ε) (y' means dy/dx), and x is on the closed unit interval. Boundary conditions: y(0,ε) = 0, y(1,ε) = 1.

We assume -0 < ε << 1.

We have two scales: "x" and "η", where η=εx is the "slow variable."

To solve, you form two solutions and recombine them later appropriately. The first solution you find by letting ε go to zero and solving the corresponding first order ODE. For the second solution, change variables to η, recompute y' and y' ', then ignore the ε term (this is equivalent to assuming x is O(1/ε)).

In general, singular perturbation occurs when the small variable (e.g. ε) is a coefficient to the highest order derivative in the ODE.

The above technique does not always work. Also, there are many ODE singular perturbation problems that look very different (need not have ε as coefficient of highest derivative). The common characteristic is that normal perturbation techniques do not work.

Reverted
The ODE example that was given (which I thought was quite good, by the way) was unnecessary in light of the article method of matched asymptotic expansions. I would recommend putting any further examples in that article. In general, this article refers to a wide class of problems and methods. As such, it should be descriptive and unifying, with links to the various methods' articles, wherein are found specific examples.

Also, the definition that recently and quite rudely replaced the one I had written, based on the literature in singular perturbation theory, was imprecise and technically inferior to mine. --Roy W. Wright 10:45, 2 May 2007 (UTC)

reply:
About definition: while the definition you give may be technically precise, it is not very readable for a layperson who has not been exposed to singular perturbations; in practice the motion of singular perturbation can only be understood by working through some examples. In addition, your definition does not cover algebraic equations or singular eigenvalue problems. Anyway I've changed to include both intuitive definition as well as your technical definition.

Also I dont think it hurts to include an ODE example here; only after going through such an example can one appreciate what is meant by singular perturbations. After all, most books on singular perturbations start with an example to explain what is meant by singular perturbation. I've partially unreverted to include it here as well, with reference to matched asymptotics. Asympt 00:20, 4 May 2007 (UTC)

is this a singular perturbation problem?
How would you solve it? All the stuff I can find on the internet shows how to solve the case where the small parameter (epsilon) is a coefficient of the highest derivative (not the case here). I don't have a lot of money or access to a large library. Also, it appears to be resistant to regular perturbation.


 * $${p^\prime}(1) = - z$$
 * $$p(\infty) = 0$$
 * $$\frac{\partial^2 p(r) }{\partial r^2} + \frac{2}{r} \frac{\partial p(r) }{\partial r} = \epsilon p(r)$$

reply: Actually, your ODE has an exact solution. Its general solution is
 * $$ p(r)= A\frac{\exp(\sqrt{\varepsilon}r) }{r} + B \frac{\exp(-\sqrt{\varepsilon}r) }{r}

$$ (as you can easily verify). So to satisfy your boundary conditions, take $$A=0, B=z \exp(\sqrt{\varepsilon})\frac{1}{\sqrt{\varepsilon}+1}$$.

Asympt 15:19, 18 August 2007 (UTC)