Alekseev–Gröbner Formula

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.

Formulation

Let $d \in \mathbb N$ be a natural number, let $T \in (0, \infty)$ be a positive real number, 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