Cauchy problem

A differential equation problem that specifies initial data on a surface or at an initial time. Its solutions must satisfy both the equation and those data.

Initial value problem: A differential equation problem that prescribes the unknown function or its derivatives at a specified starting point or time. This is the time-based form of a Cauchy problem.
Partial differential equation: An equation involving an unknown function of several variables and its partial derivatives. Many Cauchy problems concern partial differential equations with data on a surface.
Wave equation: A hyperbolic partial differential equation describing the propagation of waves through space and time. Its displacement and velocity at an initial time form standard Cauchy data.
Elliptic partial differential equation: A partial differential equation whose principal part is elliptic, as in Laplace's equation. Prescribing arbitrary data on a surface generally does not yield a well-posed Cauchy problem for elliptic equations.
Initial data: Values or functions specified at the start of a differential equation problem. These conditions distinguish a Cauchy problem from the differential equation alone.
Ordinary differential equation: A differential equation involving a function of one independent variable and its derivatives. Cauchy problems for ordinary differential equations prescribe values at an initial point.
Cauchy problem for the wave equation: The wave equation posed with initial displacement and initial velocity on a spatial slice. It is a canonical example of evolution from data at one time.
Hyperbolic partial differential equation: A partial differential equation whose characteristic structure supports finite-speed propagation of disturbances. Hyperbolic equations commonly admit well-posed evolution from suitable Cauchy data.
Existence and uniqueness theorem: A theorem establishing whether a differential equation has a solution and whether that solution is unique. Such theorems determine whether prescribed Cauchy data define one solution.
Cauchy surface: A surface that intersects every relevant causal curve exactly once in a spacetime or hyperbolic system. Its geometry determines whether initial data cover the domain of dependence.

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