“Leading Order Analysis”的概念、定义、习题解题方法及翻译-数学辞典

单词

Leading Order Analysis

释义

Leading Order Analysis

A procedure for determining the behavior of an th order Ordinary Differential Equation at a Removable Singularity without actually solving the equation. Consider

(1)

where

is Analytic in

and rational in its other arguments. Proceed by making thesubstitution

(2)

with

. For example, in the equation

(3)

making the substitution gives

(4)

The most singular terms (those with the most Negative exponents) are called the ``dominant balance terms,'' and mustbalance exponents and Coefficients at the Singularity. Here, the first two terms are dominant, so

(5)

(6)

and the solution behaves as

. The behavior in the Neighborhood of the Singularity isgiven by expansion in a Laurent Series, in this case,

(7)

Plugging this series in yields


	(8)

This gives Recurrence Relations, in this case with

arbitrary, so the

term is called the resonance or Kovalevskaya Exponent. At the resonances, the Coefficient will always bearbitrary. If no resonance term is present, the Pole present is not ordinary, and the solution must beinvestigated using a Psi Function.

See also Psi Function

References

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 330, 1989.

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