# Cauchy Problem

Also found in: Wikipedia.

## 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.

A. V. BITSADZE

References in periodicals archive ?
In the present work we discretize the reduced Cauchy problem by forward difference approximation.
In fact, due to its peculiar formulation, it leads to view the usual Einstein equations as merely initial conditions following the Cauchy problem.
Vrabie has rewritten in English the Romanian lecture notes for a course presenting a completely new approach to learning differential equations with a main emphasis on the Cauchy problem.
Previously, the elements of the matrix (5) were calculated using differencing formulas, thus the Cauchy problem was solved (that is a target function F was being calculated) for one iteration in the total number of n+1 times (n is the order of the system of equations (1)).
The subject of this paper is to study the local existence and uniqueness of solution of the Cauchy problem for the following Boussinesq-operator equation
The global existence of weak solutions to the Cauchy problem for system (1) was obtained in [9] by constructing four families of Lax-type entropies and entropy fluxes and in [10] by using a new technique from the div-curl lemma in the compensated compactness theorem.
In Section 4, we are interested in a nonlinear stochastic Cauchy problem with the white noise as initial data:
The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials.
The Cauchy problem (1) has been extensively studied in the scale of Lebesgue spaces [L.
Because differential equations which processes of real world are often non-linear and of superior order, finding an analytical solution (exact one) of Cauchy problem is difficult.
if f is Holder continuous with exponent [beta] [member of] (0, 1] and A is a sectorial operator, then the unique solution of the Cauchy problem (3.
Topics include incorporating the Lionville theorem for homogeneous elliptic differential inequalities, higher order linear parabolic equations, source of nonlinearity in the kinetic theory for active particles with focus on the formation of political opinions, existence and uniqueness solutions to a Cauchy problem modeling the dynamic of socio-political conflicts, a class of fully nonlinear PDEs from complex geometry, and a limit problem for degenerate quasilinear variational inequalities in cylinders.

Site: Follow: Share:
Open / Close