Cauchy problem

Etymology
After French mathematician.

Noun

 * 1)  For a given m-order partial differential equation, the problem of finding a solution function $$u$$ on $$\mathbb{R}^n$$ that satisfies the boundary conditions that, for a smooth manifold $$S\subset\mathbb{R}^n$$, $$\textstyle u(x) = f_0(x)$$ and $$\textstyle \frac{\partial^k u(x)}{\partial n^k} = f_k(x)$$, $$\forall x\in S$$, $$ k = 1\dots m-1$$, given specified functions $$f_k$$ defined on, and vector $$n$$ normal to, the manifold.

Usage notes
The hypersurface S is called the. The functions fk defined on S are collectively known as the of the problem.

Translations

 * Italian: problema di Cauchy