Cauchy Problem

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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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 ?
Among their topics are geometric methods for stochastic dynamical systems, an averaging principle for multi-valued stochastic differential equations driven by G-Brownian motion, Holder estimates for solutions of stochastic nonlocal diffusion equations, the Cauchy problem for a generalized Ostrovsky equation with positive dispersion, the smooth approximation of Levy processes in Skorokhod space, and error estimation on projective integration of expensive multiscale stochastic simulation.
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In the current paper, we consider the following Cauchy problem:
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