Alekseev–Gröbner Formula
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The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960 and Vladimir Mikhailovich Alekseev 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 In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives ...
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Formulation

Let d \in \mathbb N be a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
, let T \in (0, \infty) be a positive
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, and let \mu \colon , T\times \mathbb^ \to \mathbb^ \in C^( , T\times \mathbb^) be a function which is continuous on the time interval , T/math> and continuously differentiable on the d-dimensional space \mathbb^. Let X \colon , T \times \mathbb^ \to \mathbb^, (s, t, x) \mapsto X_^ be a continuous solution of the integral equation X_^ = x + \int_^ \mu(r, X_^) dr. Furthermore, let Y \in C^( , T \mathbb^) be continuously differentiable. We view Y as the unperturbed function, and X as the perturbed function. Then it holds that X_^ - Y_ = \int_^ \left( \frac X_^ \right) \left( \mu(r, Y_) - \frac Y_ \right) dr. The Alekseev–Gröbner formula allows to express the global error X_^ - Y_ in terms of the local error ( \mu(r, Y_) - \tfrac Y_) .


The Itô–Alekseev–Gröbner formula

The Itô–Alekseev–Gröbner formula is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function f \in C^(\mathbb R^, \mathbb R^) it holds that f(X_^) - f(Y_) = \int_^ f'\left( \frac X_^ \right) \frac X_^\left( \mu(r, Y_) - \frac Y_ \right) dr.


References

{{DEFAULTSORT:Alekseev-Grobner formula Nonlinear algebra Ordinary differential equations