Matched Asymptotic Expansion and stretching variables

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SUMMARY

This discussion focuses on matched asymptotic expansions in the context of ordinary differential equations (ODE) boundary value problems (BVP). The participants explore the necessity of stretching the independent variable, specifically through the transformation ##\phi = (1-x)/\epsilon##, where ##\epsilon## is a small parameter. The conversation also raises the possibility of stretching the dependent variable, exemplified by the transformation ##Y = y/\epsilon##, suggesting that such a scenario could occur under certain conditions.

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  • Understanding of ordinary differential equations (ODE)
  • Familiarity with boundary value problems (BVP)
  • Knowledge of asymptotic analysis techniques
  • Concept of boundary layers in differential equations
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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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