Cauchy problem

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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. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems. They are named after Augustin Louis Cauchy.

Suppose that the partial differential equation is defined on Rⁿ and consider a smooth manifold S ⊂ Rⁿ of dimension n − 1 (S is called the Cauchy surface). Then the Cauchy problem consists of finding the solution u of the differential equation which satisfies

$\begin{align} u(x) &= f_0(x) \qquad && \text{for all } x\in S; \\ \frac{\part^k u(x)}{\part n^k} &= f_k(x) \qquad && \text{for } k=1,\ldots,\kappa-1 \text{ and all } x\in S, \end{align}$

where $f k$ are given functions defined on the surface $S$ (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–Kovalevskaya 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.