Anfangswerte

Anfangswerte are the starting values used to determine the evolution of a system described by equations or iterative processes. The term, which comes from German (Anfang = beginning, Werte = values), is common in mathematics, physics, and numerical analysis to denote the state of the dependent variable(s) at the initial point from which a solution is generated.

In differential equations, an initial value specifies the state at a given initial time, such as y(t0) = y0. The initial value, together with the differential equation and any other required conditions, typically determines a unique solution under appropriate existence and uniqueness assumptions. In systems of differential equations, an initial vector y(t0) = y0 sets the initial state of the entire system.

In difference equations and iterative mappings, initial values seed the sequence or trajectory. For example, a0 given together with a recurrence a_{n+1} = g(a_n, n) yields the subsequent terms of the sequence.

In numerical methods, initial values are essential inputs for algorithms like Euler’s method, Runge–Kutta methods, or fixed-point iterations. The choice and accuracy of initial values affect convergence, stability, and error estimates of computed solutions.

The concept is distinct from boundary conditions, which specify constraints at multiple points or over boundaries rather than at an initial point. However, in practice initial values can influence the behavior of nonlinear or chaotic systems, where small changes in starting values may lead to divergent outcomes.

Examples include solving y' = f(y,t) with y(t0) = y0 or generating a sequence with a0 given and a_{n+1} = g(a_n, n).