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.

Introduction

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.

Formal statement

For a partial differential equation defined on Rn and a smooth manifold SRn of dimension n − 1 (S is called the Cauchy surface), the Cauchy problem consists of finding the solution u of the differential equation of order ${\displaystyle m}$ that satisfies

{\displaystyle {\begin{aligned}u(x)&=f_{0}(x)\qquad &&{\text{for all }}x\in S;\\{\frac {\partial ^{k}u(x)}{\partial n^{k}}}&=f_{k}(x)\qquad &&{\text{for }}k=1,\ldots ,m-1{\text{ and all }}x\in S,\end{aligned}}}

where ${\displaystyle f_{k}}$ are given functions defined on the surface ${\displaystyle S}$ (collectively known as the Cauchy data of the problem), and n is a normal vector to S.

Cauchy–Kowalevski theorem

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.