 Cauchy Problem
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Cauchy Problem

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that 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). It is named after Augustin Louis Cauchy.

## Formal statement

For a partial differential equation defined on Rn+1 and a smooth manifold S ? Rn+1 of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions $u_{1},\dots ,u_{N}$ of the differential equation with respect to the independent variables $t,x_{1},\dots ,x_{n}$ that satisfies

{\begin{aligned}&{\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} subject to the condition, for some value $t=t_{0}$ ,

${\frac {\partial ^{k}u_{i}}{\partial t^{k}}}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for }}k=0,1,2,\dots ,n_{i}-1$ where $\phi _{i}^{(k)}(x_{1},\dots ,x_{n})$ are given functions defined on the surface $S$ (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 $F_{i}$ are analytic in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,\phi _{j,k_{0},k_{1},\dots ,k_{n}}^{0},\dots )$ , and if all the functions $\phi _{j}^{(k)}$ are analytic in some neighborhood of the point $(x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ , then the Cauchy problem has a unique analytic solution in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ .