initialvillkoret

Initialvillkoret, or initial condition in English, is the specification of a system’s state at the start of an observation or at an initial time, typically t = 0. It is essential for determining the subsequent evolution of a model governed by differential equations, difference equations, or other dynamical rules. The number of independent initial conditions required equals the order of the governing equation; for example, a first-order ODE requires one initial value, such as y(0) = y0, while a second-order ODE requires two, such as y(0) and y'(0).

In continuous models, the initialvillkoret may specify quantities like position, velocity, temperature, or concentration at the initial time. In numerical simulations, accurate initial conditions are crucial for stability and correctness; an incorrect initialvillkoret can lead to erroneous trajectories or solutions. In partial differential equations, the initial condition describes the state on an initial time slice, often together with boundary conditions that apply on the spatial domain.

Beyond deterministic models, stochastic settings may use a random or distributed initial condition, such as an initial probability distribution for a state variable. The concept is central to the Cauchy problem, also called the initial-value problem, which contrasts with boundary conditions that apply on the domain boundary rather than at an initial time.

See also: Cauchy problem, boundary condition, well-posed problem, IVP, ODE, PDE.