Various averaging schemes for studying problems of celestial mechanics were used since works of Carl Friederich Gauss, Pierre Fatou, Boris Delone and George William Hill.
[2][3][4] 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.
[5] The method deals with differential equations in the form for a smooth function f along with appropriate initial conditions.
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
can be (approximately) removed by averaging on the right hand side of the previous equation: where