Cauchy problem

Cauchy problem

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 R"n" and consider a smooth manifold "S" ⊂ R"n" 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: egin{align}u(x) &= f_0(x) qquad && ext{for all } xin S; \frac{part^m u(x)}{part n^m} &= f_m(x) qquad && ext{for } m=1,ldots,kappa-1 ext{ and all } xin S,end{align} where f_m 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.

See also

*Cauchy boundary condition

External links

* [http://mathworld.wolfram.com/CauchyProblem.html Cauchy problem] at MathWorld.


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