# Cauchy Problem

(redirected from Cauchy problems)

## Cauchy problem

[kō·shē ‚präb·ləm]
(mathematics)
The problem of determining the solution of a system of partial differential equation of order m from the prescribed values of the solution and of its derivatives of order less than m on a given surface.

## Cauchy Problem

one of the fundamental problems of the theory of differential equations, first studied systematically by A. Cauchy. It consists in finding a solution u(x, t), for x = (xi, …, xn), of a differential equation of the form

satisfying the initial conditions

where G0—the carrier of the initial data—is a region in the hyperplane t = t0 of the space of variables x1, …, xn. When F and fk, for k = 0, …, m — 1, are analytic functions of their arguments, then the Cauchy problem (1), (2) always has a unique solution in some region G of the space of variables t, x containing G0. This solution, however, can prove to be unstable (that is, a small change in the initial data can cause a large change in the solution), for example, for cases when equation (1) is elliptic. If equation (1) is not hyperbolic and the initial data are not analytic, then the Cauchy problem (1), (2) can lose meaning.

### REFERENCES

Courant, R., and D. Hilbert. Metody matematicheskoi fiziki, vol. 2. Moscow-Leningrad, 1951. (Translated from German.)
Tikhonov, A. N., and A. A. Samarskii. Uraveniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.

References in periodicals archive ?
Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity.
The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials.
For example, the Cauchy problems on differential equations
i]), 1 [less than or equal to] i [less than or equal to] l and the Cauchy problem
An energy regularization for Cauchy problems of Laplace equation in annulus domain.
This problem is often refered to as Cauchy problem and it is well known that these problems are ill-posed [2, 3].
1999, Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain, Ann.
2001, Existence and uniqueness of pseudo almost periodic solutions of semilinear Cauchy problems with nondense domain, Nonlinear Analysis, Vol.
They start with the basics of data processing, including interpolating data, Monte Carlo methods and spectrum analysis, followed by inverse problems (such as linear reduction), Cauchy problems in ordinary differential equations, nonlinear dynamics (including dynamical systems and their attractors, bifurcations including Hopf bifurcations and strange attractors), boundary-value problems (including the Euler method) and partial differential equations, including problem formulation, artificial diffusion and a range of examples.
These proceedings from the August 2003 workshop include written versions of formal presentations, with general topics including black hole physics, Cauchy problems, current issues in cosmology, Einstein- Yang-Mills (and Higgs) systems and analytic and perturbation models.
They then systematically study the qualitative behavior of the solution semigroup and then extend the theory to second-order Cauchy problems with delay, parabolic problems with delays in the highest order derivatives, and other problems.

Site: Follow: Share:
Open / Close