Differenssiyhtälö

Differenssiyhtälö is the Finnish term for a difference equation. A difference equation is a relation involving values of a function at different points, where the points are separated by a constant difference. These equations are the discrete-time analogue of differential equations, which involve continuous variables.

Difference equations are used to model phenomena that change over discrete time steps. Examples include population growth, financial models, and signal processing. The independent variable in a difference equation is typically an integer, representing time steps. The dependent variable is the function whose values are related at these discrete points.

The order of a difference equation is determined by the largest difference between the arguments of the function. For instance, a first-order difference equation relates the value of a function at time $n$ to its value at time $n-1$. A second-order equation would involve values at $n$, $n-1$, and $n-2$.

Solving a difference equation means finding a function that satisfies the given relation. Similar to differential equations, there can be general solutions and particular solutions, often determined by initial conditions. Techniques for solving them include iteration, characteristic equations for linear homogeneous equations, and methods for handling non-homogeneous or non-linear cases. The study of difference equations is fundamental in discrete mathematics and various applied fields.