Cauchy problem

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.

See also

• Cauchy boundary condition

References

• Hadamard, Jacques (2003) [1923], Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Phoenix editions, New York: Dover Publications, ISBN 0486495493, MRJFM [http://www.zentralblatt-math.org/zmath/en/search/?q=an:49.0725.04&format=complete 49.0725.04 0051411 [[Jahrbuch über die Fortschritte der Mathematik|JFM]] [http://www.zentralblatt-math.org/zmath/en/search/?q=an:49.0725.04&format=complete 49.0725.04]], http://books.google.com/books?id=B25O-x21uqkC 

External links

• Cauchy problem at MathWorld.