The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear
variation of constants
In mathematics, variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations.
For first-order inhomogeneous linear differential equations it is usually possible ...
formula which was proven independently by
Wolfgang Gröbner
Wolfgang Gröbner (11 February 1899 – 20 August 1980) was an Austrian mathematician. His name is best known for the Gröbner basis, used for computations in algebraic geometry. However, the theory of Gröbner bases for polynomial rings was dev ...
in 1960 and
Vladimir Mikhailovich Alekseev
Vladimir Mikhailovich Alekseev (Владимир Михайлович Алексеев, sometimes transliterated as "Alexeyev" or "Alexeev", 17 June 1932, Bykovo, Ramensky District, Moscow Oblast – 1 December 1980) was a Russian mathematician who ...
in 1961. It expresses the global error of a perturbation in terms of the local error and has many applications for studying perturbations of
ordinary differential equations.
Formulation
Let
be a natural number, let
be a positive real number, and let
be a function which is continuous on the time interval