Talk:Method of dominant balance

Confusing assumption in example

Hi,

I find the way the example ODE is treated rather confusing. The assumption $S_{0}^{'}\sim (S_{0}^{'})^{2}$ comes out of nowhere. Why not simply drop the terms which go as $1/x$ given that we are investigating the limit $x\rightarrow \infty$? Then, the ODE becomes separable once on changes variables to $T=S^{'}$. Would this not be a better way to illustrate the method?

Please sign your messages with 4 tildes. I agree. Dominant balance means you identify 2 dominant terms in an equation solve this simplified equation of 2 terms, substitute the solution into the full equation, and determine the solution's error. This determines the first term of the solution. Repeat to obtain the second term etc. With each term the error becomes asymptotically smaller. You solve for small x near 0 to get the inner solution. Then solve for large x to get the outer solution. This is called the boundary layer approach. The reasoning in this article is unclear. I will fix this when I have time. TMM53 (talk) 18:18, 16 April 2024 (UTC)

Proposed revision

This is a proposed revision. Any suggestions, please send me a message and let us discuss. I will revise this document in the near future.

The reasons for the revision are the following:

• The revision describes dominant balance method as an algorithm rather than a heuristic approach. In the revised document examples, the same algorithmic steps are repeated and reliably give the correct result. In the current document's differential equation example, the reader is asked to make many informed guesses or assumptions. The reason why these assumptions are made is unclear to the reader.
• The revision introduces, defines and uses the asymptotic notation (${\textstyle \gg ,\ll ,\sim }$, o()) in a precise way. A reader unaware of asymptotic notation would find the current document confusing. Also, in each instance when asymptotic notation is used, the limit should be given to make it easier for the reader. Also, the current document provides no source or internal links for the reader to understand asymptotic notations.
• Some statements in the current document are not correct. The substitution ${\textstyle y=e^{S(x)}}$ is done only in some applications of dominant balance and not in every instance of when the method is used. Also, dominant balance is not limited to ordinary differential equations.
• The revision provides a source for the differential equation example. This is not done in the current document.
• The revision adds relevant content and many references.

Proposed revision

• User:TMM53/Method of dominant balance-2024-04-17

Thanks TMM53 (talk) TMM53 (talk) 07:25, 6 May 2024 (UTC)