Initial-value problem

An initial-value problem asks for a solution to a differential equation subject to conditions that specify the unknown function or fields at an initial time.

Differential equation: An equation relating an unknown function to one or more of its derivatives. It supplies the governing relation that the initial conditions constrain.
Existence and uniqueness theorem: A theorem giving conditions under which a differential equation has a solution and that solution is unique. It states when the prescribed initial data determine a local solution.
Cauchy problem: A PDE problem specifying initial data on a hypersurface, often together with equations governing evolution away from it. It is the standard initial-data formulation for many partial differential equations.
Boundary-value problem: A differential-equation problem whose conditions are imposed at boundaries of the domain. It prescribes constraints across a domain instead of specifying a starting state.
Global existence: The property that a solution exists throughout the full time or domain of interest. Local solvability does not guarantee an initial-value solution persists indefinitely.
Initial condition: A condition specifying a function or its derivatives at a designated initial point or time. It provides the data that distinguish the desired solution from other solutions.
Picard–Lindelöf theorem: A theorem ensuring local existence and uniqueness for an ODE with a locally Lipschitz right-hand side. It provides a standard test for whether an ODE initial-value problem is locally well posed.
Wave equation: A second-order PDE describing propagation of waves through space and time. Its evolution is commonly determined by initial displacement and velocity.
Two-point boundary-value problem: A boundary-value problem with conditions imposed at two distinct points of an interval. Its endpoint constraints generally cannot be treated as initial data at one endpoint.
Navier–Stokes existence and smoothness: The open mathematical problem of proving global smooth solutions for three-dimensional incompressible Navier–Stokes equations. It asks whether smooth initial velocity data always yield smooth solutions for all time.

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