Cauchy Problem
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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.[1] A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin Louis Cauchy.

Formal statement

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[2]

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.

Cauchy-Kowalevski theorem

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 .

See also

References

  1. ^ Jacques Hadamard (1923), Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Phoenix editions
  2. ^ Petrovskii, I. G. (1954). Lectures on partial differential equations. Interscience Publishers, Inc, Translated by A. Shenitzer, (Dover publications, 1991)

External links


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