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The '''Krylov–Bogolyubov averaging method''' ('''Krylov–Bogolyubov method of averaging''') is a mathematical method for approximate analysis of oscillating processes in non-linear mechanics.<ref>[http://eom.springer.de/K/k055940.htm Krylov–Bogolyubov method of averaging] at Springer Encyclopaedia of Mathematics</ref> The method is based on the averaging principle when the exact differential equation of the motion is replaced by its averaged version. The method is named after [[Nikolay Mitrofanovich Krylov|Nikolay Krylov]] and [[Nikolay Bogoliubov]].


Various averaging schemes for studying problems of celestial mechanics were used since works of [[Carl Friedrich Gauss|Gauss]], [[Pierre Fatou|Fatou]], Delone, [[George William Hill|Hill]]. The importance of the contribution of Krylov and Bogoliubov is that they developed a general averaging approach and proved that the solution of the averaged system approximates the exact dynamics.<ref>{{cite book |title=Methodes approchees de la mecanique non-lineaire dans leurs application a l'Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s'y rapportant|last=N. M. Krylov|authorlink=|coauthors=N. N. Bogolyubov|year=1935|publisher=Academie des Sciences d'Ukraine|location=Kiev|isbn=|pages=|language=French}}</ref><ref>{{cite book |title=Introduction to non-linear mechanics|last= N. M. Krylov|authorlink= |coauthors=N. N. Bogolyubov|year=1937|language=Russian |publisher=Izd-vo AN SSSR|location=Kiev|isbn=|pages=}}</ref><ref>{{cite book |title=Introduction to non-linear mechanics|last= N. M. Krylov|authorlink= |coauthors=N. N. Bogolyubov|year=1947 |publisher=Princeton Univ. Press|location=Princeton|isbn=9780691079851|pages=|language=English, partial translation from Russian}}</ref>


==Background==
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Krylov-Bogoliubov averaging can be used to approximate oscillatory problems when a classical perturbation expansion fails. That is [[singular perturbation]] problems of oscillatory type, for example Einstein's correction to the [[Two-body problem in general relativity|perihelion precession of Mercury]].<ref name="Smith">{{cite book | last = Smith | first = Donald | title = Singular-Perturbation Theory | publisher = Cambridge University Press | location = Cambridge | year = 1985 | isbn = 0-521-30042-8 }}</ref>
 
==Derivation==
The method deals with differential equations in the form
 
:<math>
\frac{d^2u}{dt^2} + k^2 u = a + \varepsilon f\left(u,\frac{du}{dt}\right)
</math>
for a smooth function ''f'' along with appropriate initial conditions. The parameter ''ε'' is assumed to satisfy
 
:<math>
0 < \varepsilon \ll k.
</math>
If ''ε''&nbsp;=&nbsp;0 then the equation becomes that of the simple harmonic oscillator with constant forcing, and the general solution is
 
:<math>
u(t) = \frac{a}{k^2} + A \sin (kt + B),
</math>
where ''A'' and ''B'' are chosen to match the initial conditions. The solution to the perturbed equation (when ''ε''&nbsp;≠&nbsp;0) is assumed to take the same
form, but now ''A'' and ''B'' are allowed to vary with ''t'' (and&nbsp;''ε''). If it is also assumed that
:<math>
\frac{du}{dt} = kA(t) \cos (kt + B(t)),
</math>
then it can be shown that ''A'' and ''B'' satisfy the differential equation:<ref name="Smith" />
:<math>
\frac{d}{dt} \begin{bmatrix} A \\ B \end{bmatrix}  = \frac{\varepsilon}{k} f\left( \frac{a}{k^2} + A \sin (\phi), kA \cos (\phi)\right) \begin{bmatrix} \cos(\phi)  \\ - \frac{1}{A} \sin(\phi)  \end{bmatrix},
</math>
where <math> \phi = kt + B </math>. Note that this equation is still exact — no approximation has been made as yet. The method of Krylov and Bogolyubov is to note that the functions A and B vary slowly
with time (in proportion to ε), so their dependence on &phi; can be (approximately) removed by averaging on the right hand side of the previous equation:
:<math>
\frac{d}{dt} \begin{bmatrix} A_0 \\ B_0 \end{bmatrix}  = \frac{\varepsilon}{2\pi k} \int_0^{2 \pi} f( \frac{a}{k^2} + A \sin (\theta), kA \cos (\theta)) \begin{bmatrix} \cos(\theta)  \\ - \frac{1}{A_0} \sin(\theta)  \end{bmatrix} d\theta,
</math>
where <math>A_0</math> and <math>B_0</math> are held fixed during the integration. After solving this (possibly) simpler set of differential equations, the Krylov–Bogolyubov averaged approximation for the original function is then given by
:<math>
u_0(t,\varepsilon) := \frac{a}{k^2} + A_0(t,\varepsilon) \sin (kt + B_0(t,\varepsilon)).
</math>
This approximation has been shown to satisfy <ref>{{cite book | last = Bogolubov | first = N. | title = Asymptotic Methods in the Theory of Non-Linear Oscillations | publisher = Gordon & Breach | location = Paris | year = 1961 | isbn = 978-0-677-20050-7 }}</ref>
:<math>
\left| u(t,\varepsilon) - u_0(t,\varepsilon) \right| \le C_1 \varepsilon,
</math>
where t satisfies
:<math>
0 \le t \le \frac{C_2}{\varepsilon}
</math>
for some constants <math>C_1</math> and <math>C_2</math>, independent of ε.
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Krylov-Bogoliubov averaging method}}
[[Category:Dynamical systems]]

Latest revision as of 15:32, 8 January 2015


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