Cauchy problem

From Wikipedia, the free encyclopedia

You have new messages (last change).

Consider a smooth hypersurface S\in\mathbb{R}^n having a continuous, non-tangential direction field described by unitary vectors \hat{\mathbf{d}(\mathbf{x})},\mathbf{x}\in S, i.e.

\hat{\mathbf{d}(\mathbf{x})}\cdot\hat{\mathbf{n}(\mathbf{x})}\neq0, \forall\mathbf{x}\in S

where \hat{\mathbf{n}(\mathbf{x})} is the unitary vector perpendicular to S.

The general Cauchy problem, consists on finding the solution u of a κ − order differential equation that also satisfies the conditions:

u(\mathbf{x})=f_0(\mathbf{x}), \mathbf{x}\in S
\frac{\part^m u(\mathbf{x})}{\part d^m}=f_m(\mathbf{x}), m=1,\ldots,\kappa-1

where fm are given functions defined on surface S (Cauchy surface).

[edit] See also

[edit] External links

This mathematical analysis-related article is a stub. You can help Wikipedia by expanding it.
Retrieved from "http://en.wikipedia.org../../../c/a/u/Cauchy_problem.html"