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) or it can be either of them. It is named after Augustin Louis Cauchy.
For a partial differential equation defined on Rn+1 and a smooth manifold S ⊂ Rn+1 of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions of the differential equation with respect to the independent variables that satisfies
subject to the condition, for some value ,
where are given functions defined on the surface (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.
The Cauchy–Kowalevski theorem states that If all the functions are analytic in some neighborhood of the point , and if all the functions are analytic in some neighborhood of the point , then the Cauchy problem has a unique analytic solution in some neighborhood of the point .
- Jacques Hadamard (1923), Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Phoenix editions
- Petrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991)