Differensligninger

Differensligning, or a difference equation, is an equation that defines a sequence by relating its terms. It is the discrete-time counterpart to a differential equation and appears in mathematics, computer science, and various applications such as population dynamics, economics, and signal processing.

Difference equations are commonly classified by order (first-order, second-order, or higher), by linearity (linear versus nonlinear), by coefficients (constant versus variable), and by homogeneity (homogeneous with no independent term versus non-homogeneous). They can also be described as autonomous if the rule uses only previous terms and constants, or non-autonomous if the index n appears explicitly.

Solving difference equations involves several methods. One can iterate from given initial values to generate terms. For linear constant-coefficient equations, solutions often come from the characteristic equation, producing a general solution that is a linear combination of terms like n^k r^n. Non-homogeneous equations require a particular solution in addition to the general solution of the corresponding homogeneous equation. Other powerful techniques include generating functions and the Z-transform, especially for sequences arising in engineering and computer science.

Common examples illustrate the ideas. A first-order linear autonomous equation a_{n+1} = r a_n has solution a_n = a_0 r^n. A first-order non-homogeneous form a_{n+1} = r a_n + b yields a_n = a_0 r^n + b (r^n - 1)/(r - 1) for r ≠ 1 (and a_n = a_0 + n b when r = 1). Nonlinear examples include the logistic map x_{n+1} = r x_n (1 - x_n).

Applications include modeling populations, financial processes, and digital signals. Difference equations are discrete analogues to differential equations, and they often arise from discretizing continuous models or from inherently discrete phenomena.