Cauchy problem
Encyclopedia
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 problem
s and are to be contrasted with boundary value problem
s. 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
where
are given functions defined on the surface 
(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.
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
that satisfies certain conditions which are given on a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
in the domain. Cauchy problems are an extension of initial value problem
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
s and are to be contrasted with boundary value problem
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
s. They are named after Augustin Louis Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
.
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
Cauchy surface
Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future uniquely....
). Then the Cauchy problem consists of finding the solution u of the differential equation which satisfies
where
are given functions defined on the surface (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.
External links
- Cauchy problem at MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
.