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.

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.

For a partial differential equation defined on Rⁿ and a smooth manifold S ⊂ Rⁿ 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

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.

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