View source for Cauchy problem

{{Short description|Class of problems for PDEs}}
{{Differential equations}}
A '''Cauchy problem''' in mathematics asks for the solution of a [[Partial differential equation|partial differential equation]] that satisfies certain conditions that are given on a [[Hypersurface|hypersurface]] in the domain.<ref>{{cite book |first=Jacques |last=Hadamard |year=1923 |title=Lectures on Cauchy's Problem in Linear Partial Differential Equations |location=New Haven |publisher=Yale University Press |pages=4–5 |oclc=1880147 }}</ref> A Cauchy problem can be an [[Initial value problem|initial value problem]] or a [[Boundary value problem|boundary value problem]] (for this case see also [[Cauchy boundary condition]]). It is named after [[Biography:Augustin-Louis Cauchy|Augustin-Louis Cauchy]].

==Formal statement==

For a partial differential equation defined on '''R'''<sup>''n+1''</sup> and a smooth manifold ''S'' ⊂ '''R'''<sup>''n+1''</sup> of dimension ''n'' (''S'' is called the [[Cauchy surface]]), the Cauchy problem consists of finding the unknown functions <math>u_1,\dots,u_N</math> of the differential equation with respect to the independent variables <math>t,x_1,\dots,x_n</math> that satisfies<ref>{{cite book |last=Petrovsky |first=I. G. |orig-year=1954 |title=Lectures on Partial Differential Equations |publisher=Interscience |location=New York |translator-first=A. |translator-last=Shenitzer |edition=Dover |year=1991 |isbn=0-486-66902-5 }}</ref>
<math display="block">\begin{align}&\frac{\partial^{n_i}u_i}{\partial t^{n_i}} = F_i\left(t,x_1,\dots,x_n,u_1,\dots,u_N,\dots,\frac{\partial^k u_j}{\partial t^{k_0}\partial x_1^{k_1}\dots\partial x_n^{k_n}},\dots\right) \\
&\text{for } i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0<n_j
\end{align}</math>
subject to the condition, for some value <math>t=t_0</math>,

<math display="block">\frac{\partial^k u_i}{\partial t^k}=\phi_i^{(k)}(x_1,\dots,x_n)
\quad \text{for } k=0,1,2,\dots,n_i-1</math>

where <math>\phi_i^{(k)}(x_1,\dots,x_n)</math> are given functions defined on the surface <math>S</math> (collectively known as the '''Cauchy data''' of the problem). The derivative of order zero means that the function itself is specified.

==Cauchy–Kowalevski theorem==
The [[Cauchy–Kowalevski theorem]] states that ''If all the functions <math>F_i</math> are [[Analytic function|analytic]] in some neighborhood of the point <math>(t^0,x_1^0,x_2^0,\dots,\phi_{j,k_0,k_1,\dots,k_n}^0,\dots)</math>, and if all the functions <math>\phi_j^{(k)}</math> are analytic in some neighborhood of the point <math>(x_1^0,x_2^0,\dots,x_n^0)</math>, then the Cauchy problem has a unique analytic solution in some neighborhood of the point <math>(t^0,x_1^0,x_2^0,\dots,x_n^0)</math>''.

== See also ==
*[[Cauchy boundary condition]]
*[[Physics:Cauchy horizon|Cauchy horizon]]

==References==
{{reflist}}

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


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

de:Anfangswertproblem#Partielle Differentialgleichungen

{{Sourceattribution|Cauchy problem}}