# 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 ?
Let us consider finite difference approximation of the Cauchy problems of nonlinear partial differential equations (PDE's) of the normal form, and we show here its convergence independently of stability of the Cauchy problems.
FBV Study some examples of Cauchy problems and variational problems with free boundary values by exploiting the geometric structures on the spaces of isotropic flags and non-maximal isotropic elements of a meta-symplectic space, in continuity with the Applicant's own work.
In the third of six volumes on generalized functions, Gel'fand and Silov apply the apparatus of generalized functions to investigate the problems of determining uniqueness and correctness classes for solutions of the Cauchy problems for systems with constant (or only time-dependent) coefficients, and the problem of eigenfunctions expansions for self-adjoining differential operators.
Next, the Caputo-Hadamard fractional derivative which is a Caputo-type modification of the Hadamard fractional derivative is suggested in [18], some fundamental theorems of this fractional derivative were proved in [19], and Cauchy problems of a differential equation with a left Caputo-Hadamard fractional derivative were studied in spaces of continuously differentiable functions in [20].
Here, to study some nonlinear stochastic Cauchy problems, we choose to reformulate them correctly in the framework of the (C, E, P)-algebras of Marti [13] in order to show that, following the example of the theory of Colombeau [4, 5], these algebras may serve as a tool for treating singular processes in stochastic analysis.
For example, the Cauchy problems on differential equations
Then we associate with (1), (2) a net of Cauchy problems for ordinary functional differential equations.
McKibben, Existence results for a class of abstract nonlocal Cauchy problems, Nonlinear Anal.
An energy regularization for Cauchy problems of Laplace equation in annulus domain.
Hence, we obtain the following system of Cauchy problems of ordinary differential equations
1999, Composition of pseudo almost periodic functions and Cauchy problems with operator of nondense domain, Ann.

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