Krylov–Bogoliubov averaging method

Background

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 perihelion precession of Mercury.

Derivation

The method deals with differential equations in the form

: \frac{d^2u}{dt^2} + k^2 u = a + \varepsilon f\left(u,\frac{du}{dt}\right) for a smooth function f along with appropriate initial conditions. The parameter ε is assumed to satisfy

: 0 If ε = 0 then the equation becomes that of the simple harmonic oscillator with constant forcing, and the general solution is

: u(t) = \frac{a}{k^2} + A \sin (kt + B), where A and B are chosen to match the initial conditions. The solution to the perturbed equation (when ε ≠ 0) is assumed to take the same form, but now A and B are allowed to vary with t (and ε). If it is also assumed that : \frac{du}{dt} = kA(t) \cos (kt + B(t)), then it can be shown that A and B satisfy the differential equation: : \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}, where \phi = kt + B . 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: : \frac{d}{dt} \begin{bmatrix} A_0 \ B_0 \end{bmatrix} = \frac{\varepsilon}{2\pi k} \int_0^{2 \pi} f\left( \frac{a}{k^2} + A_0 \sin (\theta), kA_0\cos (\theta)\right) \begin{bmatrix} \cos(\theta) \ - \frac{1}{A_0} \sin(\theta) \end{bmatrix} d\theta, where A_0 and B_0 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 : u_0(t,\varepsilon) := \frac{a}{k^2} + A_0(t,\varepsilon) \sin (kt + B_0(t,\varepsilon)). This approximation has been shown to satisfy : \left| u(t,\varepsilon) - u_0(t,\varepsilon) \right| \le C_1 \varepsilon, where t satisfies : 0 \le t \le \frac{C_2}{\varepsilon} for some constants C_1 and C_2, independent of ε.

References

References

1. [https://encyclopediaofmath.org/wiki/Krylov-Bogolyubov_method_of_averaging Krylov–Bogolyubov method of averaging] at [[Encyclopedia of Mathematics]]
2. N. M. Krylov. (1935). "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". Académie des Sciences d'Ukraine.
3. N. M. Krylov. (1937). "Introduction to non-linear mechanics". Izd-vo AN SSSR.
4. N. M. Krylov. (1947). "Introduction to non-linear mechanics". Princeton Univ. Press.
5. Smith, Donald. (1985). "Singular-Perturbation Theory". Cambridge University Press.
6. Bogoliubov, N.. (1961). "Asymptotic Methods in the Theory of Non-Linear Oscillations". Gordon & Breach.