Talk:Singular perturbation

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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)

[edit] 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.