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:
“Too poor for a bribe, and too proud to importune,
He had not the method of making a fortune.”
—Thomas Gray (1716–1771)
“The insidiousness of science lies in its claim to be not a subject, but a method. You could ignore a subject; no subject is all-inclusive. But a method can plausibly be applied to anything within the field of consciousness.”
—Katharine Fullerton Gerould (1879–1944)
“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)
““What I want is a strange conjunction with you—” he said quietly; “Mnot meeting and mingling;Myou are quite right:Mbut an equilibrium, a pure balance of two single beings:Mas the stars balance each other.””
—D.H. (David Herbert)