# Cauchy problem

## From Wikipedia, the free encyclopedia

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.^{[1]} A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition) or it can be either of them. It is named after Augustin Louis Cauchy.

## Formal statement

For a partial differential equation defined on **R**^{n+1} and a smooth manifold *S* ⊂ **R**^{n+1} of dimension *n* (*S* is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions of the differential equation with respect to the independent variables that satisfies^{[2]}

subject to the condition, for some value ,

where are given functions defined on the surface (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 are analytic in some neighborhood of the point , and if all the functions are analytic in some neighborhood of the point , then the Cauchy problem has a unique analytic solution in some neighborhood of the point *.

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