Matched Asymptotic Expansion and stretching variables

In summary, Matched Asymptotic Expansion and stretching variables are techniques used in asymptotic analysis to simplify complex mathematical problems. Matched Asymptotic Expansion involves using a series of approximations to find a solution to a problem, while stretching variables involves changing the variables in a problem to make it easier to solve. Both techniques are commonly used in engineering and physics to solve problems that involve large or small parameters. By using these techniques, researchers can obtain accurate solutions to difficult problems that would otherwise be impossible to solve.
  • #1
member 428835
Hi PF!

Regarding matched asymptotic expansions, given an ODE BVP, I have learned a boundary layer can arise, where we need to stretch the independent variable through carefully selection i.e. if ##x## is the independent variable, perhaps ##\phi = (1-x)/\epsilon : \epsilon \ll 1##.

Would we ever see a situation where we would have to stretch the dependent variable, say the ODE was over ##y(x)##. Something perhaps like ##Y = y/\epsilon : \epsilon \ll 1##, as an example.
 
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  • #2
Nevermind, I think it's possible!
 

1. What is a matched asymptotic expansion?

A matched asymptotic expansion is a mathematical technique used to approximate the solution of a problem that has multiple scales. It involves breaking down the problem into different regions, or scales, and matching solutions in these regions to create an overall solution that is valid in all scales.

2. How does stretching variables relate to matched asymptotic expansion?

Stretching variables are used in matched asymptotic expansion to transform the original problem into a new problem with simpler boundary conditions. This makes it easier to match solutions in different regions and obtain a more accurate overall solution.

3. Can matched asymptotic expansion be used for nonlinear problems?

Yes, matched asymptotic expansion can be used for both linear and nonlinear problems. However, for nonlinear problems, the matching of solutions in different regions can be more challenging and may require additional techniques.

4. What are the limitations of matched asymptotic expansion?

One limitation of matched asymptotic expansion is that it is only valid in the regions where the solution is matched. This means that it may not accurately capture the behavior of the solution in regions where the matching is not performed. Additionally, it is a perturbation method and may not provide accurate solutions for problems with large perturbations.

5. How is matched asymptotic expansion different from other perturbation techniques?

Matched asymptotic expansion differs from other perturbation techniques in that it considers multiple scales in the problem and matches solutions in these scales to obtain an overall solution. This makes it more accurate in capturing the behavior of the solution in different regions compared to other perturbation methods.

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