Method of Dominant Balance
The method of dominant balance is used to determine the asymptotic behavior of solutions to an ODE without solving it. The process is iterative in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.
The process is as follows:
1. Assume that the asymptotic behavior has the form
-
- .
2. Make a clever guess as to which terms in the ODE may be negligible in the limit we are interested in.
3. Drop those terms and solve the resulting ODE.
4. Check that the solution is consistent with step 2. If this is the case, then we have the controlling factor of the asymptotic behavior. Otherwise, we need to try dropping different terms in step 2.
5. Repeat the process using our result as the first term of the solution.
Read more about this topic: Asymptotic Analysis
Famous quotes containing the words method of, method, dominant and/or balance:
“Methinks the human method of expression by sound of tongue is very elementary, & ought to be substituted for some ingenious invention which should be able to give vent to at least six coherent sentences at once.”
—Virginia Woolf (1882–1941)
“... the one lesson in the ultimate triumph of any great actress has been to enforce the fact that a method all technique or a method all throes, is either one or the other inadequate, and often likely to work out in close proximity to the ludicrous.”
—Mrs. Leslie Carter (1862–1937)
“In the dominant Western religious system, the love of God is essentially the same as the belief in God, in God’s existence, God’s justice, God’s love. The love of God is essentially a thought experience. In the Eastern religions and in mysticism, the love of God is an intense feeling experience of oneness, inseparably linked with the expression of this love in every act of living.”
—Erich Fromm (1900–1980)
“True balance requires assigning realistic performance expectations to each of our roles. True balance requires us to acknowledge that our performance in some areas is more important than in others. True balance demands that we determine what accomplishments give us honest satisfaction as well as what failures cause us intolerable grief.”
—Melinda M. Marshall (20th century)