Cauchy problem

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The Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain side conditions which are given on a hypersurface in the domain. It is an extension of the initial value problem.

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^m u(x)}{\part n^m} &= f_m(x) \qquad && \text{for } m=1,\ldots,\kappa-1 \text{ and all } x\in S, \end{align}

where fm are given functions defined on the surface S, 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 a unique solutions under certain conditions.

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