en>ChrisGualtieri: General Fixes, added Empty section (1) tag using AWB

2013-11-08T01:40:58Z

General Fixes, added Empty section (1) tag using AWB

← Older revision		Revision as of 03:40, 8 November 2013
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	~~Petroleum Engineer Stewart from Quebec City, has numerous pursuits~~ which ~~include motorbikes~~, ~~diet~~ and ~~walking~~. ~~Likes to visit unfamiliar locations like Al~~-~~Hijr Archaeological Site~~ (~~Madâin Sâlih~~).		A '''Cauchy problem''' in mathematics asks for the solution of a [[partial differential equation]] that satisfies certain conditions which are given on a [[hypersurface]] in the domain. A Cauchy problem can be an [[initial value problem]] or a [[boundary value problem]] (for this case see also [[Cauchy boundary condition]]), but it can be none of them. They are named after [[Augustin Louis Cauchy]].

			Suppose that the partial differential equation is defined on '''R'''<sup>''n''</sup> and consider a [[smooth manifold]] ''S'' ⊂ '''R'''<sup>''n''</sup> of dimension ''n'' − 1 (''S'' is called the [[Cauchy surface]]). Then the Cauchy problem consists of finding the solution ''u'' of the differential equation that satisfies
			:<math> \begin{align}
			u(x) &= f_0(x) \qquad && \text{for all } x\in S; \\
			\frac{\part^k u(x)}{\part x^k} &= f_k(x) \qquad && \text{for } k=1,\ldots,\kappa-1 \text{ and all } x\in S,
			\end{align} </math>
			where <math>f_k</math> are given functions defined on the surface <math>S</math> (collectively known as the '''Cauchy data''' of the problem), ''n'' is a [[normal vector]] to ''S'', and κ denotes the order of the differential equation.

			The [[Cauchy–Kowalevski theorem]] says that Cauchy problems have unique solutions under certain conditions, the most important of which being that the Cauchy data and the coefficients of the partial differential equation be [[real analytic]] functions.

			== See also ==
			*[[Cauchy boundary condition]]

			==References==
			*{{citation\|title=Lectures on Cauchy's Problem in Linear Partial Differential Equations\|first=Jacques\|last= Hadamard\|year=2003\|origyear= 1923\|isbn=0-486-49549-3\|series=Dover Phoenix editions\|url=http://books.google.com/books?id=B25O-x21uqkC\|mr=0051411 {{JFM\|49.0725.04}}\|publisher=Dover Publications\|place= New York}}

			== External links ==
			* [http://mathworld.wolfram.com/CauchyProblem.html Cauchy problem] at [[MathWorld]].

			[[Category:Partial differential equations]]
			[[Category:Mathematical problems]]

			{{Mathanalysis-stub}}

			[[de:Anfangswertproblem#Partielle_Differentialgleichungen]]

en>EmausBot: r2.7.3) (Robot: Adding de:Mathieusche Differentialgleichung

2012-09-02T11:14:30Z

r2.7.3) (Robot: Adding de:Mathieusche Differentialgleichung

New page

Petroleum Engineer Stewart from Quebec City, has numerous pursuits which include motorbikes, diet and walking. Likes to visit unfamiliar locations like Al-Hijr Archaeological Site (Madâin Sâlih).

Mathieu function - Revision history

en>ChrisGualtieri: General Fixes, added Empty section (1) tag using AWB

en>EmausBot: r2.7.3) (Robot: Adding de:Mathieusche Differentialgleichung