Methods of Mathematical Modelling by Thomas Witelski & Mark Bowen

Author:Thomas Witelski & Mark Bowen
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(ii)The leading order coefficient is finite, as (i.e. we are looking for regular solutions; we tackle singular solutions later).

Substituting the leading order term into (6.13) yields

(6.19)

In order for this equation to hold as , at least two of the terms must balance in asymptotic scales, with the remaining terms being sub-dominant (i.e. asymptotically smaller than the retained terms as ). The smaller terms can then be neglected in determining the leading order solution. The set of dominant terms that balance to yield the leading order solutions are called the dominant terms and this argument is referred to as the principle of dominant balance.

Ignoring all O(1) coefficients for the time-being, consider the three possibilities for balancing the asymptotic gauges of the potential dominant terms in (6.19):

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