Initial condition

An initial condition specifies a solution’s value or state at a chosen starting point. Together with a differential equation, it can determine a particular solution.

Initial value problem: A differential equation problem that specifies the unknown function or its derivatives at an initial point. It pairs a differential equation with initial data to seek a particular solution.
Differential equation: An equation relating an unknown function to one or more of its derivatives. The equation supplies the rule that initial data help solve.
Cauchy momentum equation: A continuum-mechanics equation expressing conservation of linear momentum in a deformable body. Predicting motion from this equation requires specifying the body's initial velocity.
Boundary value problem: A differential equation problem whose solution must satisfy conditions imposed at boundary locations. It constrains a solution at domain boundaries rather than prescribing its starting state.
Ordinary differential equation: A differential equation involving a function of one independent variable and its ordinary derivatives. Initial conditions commonly select solutions to equations in this broader class.
Solution of a differential equation: A function that satisfies a differential equation on a specified domain. An initial condition distinguishes one candidate solution from others.
Heat equation: A partial differential equation describing how temperature diffuses through a medium. Its time-dependent solutions use an initial temperature distribution.
Dirichlet boundary condition: A boundary condition that specifies the value of a function on the boundary of its domain. It prescribes function values by position on a boundary, unlike data at an initial point.
Existence and uniqueness theorem: A result establishing when a differential equation has a solution and whether that solution is unique. It states conditions under which specified initial data determine exactly one solution.
Initial value: The value of an unknown function specified at the initial point of a problem. It is the simplest form of prescribed initial data.

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