difference equation

Equation involving differences between successive values of a function of a discrete variable (i.e., one defined for a sequence of values that differ by the same amount, usually 1).

A function of such a variable is a rule for assigning values in sequence to it. For example, f(x + 1) = xf(x) is a difference equation. Methods developed for solving such equations have much in common with methods for solving linear differential equations, which difference equations are often used to approximate.

* * *

      mathematical equality involving the differences between successive values of a function of a discrete variable. A discrete variable is one that is defined or of interest only for values that differ by some finite amount, usually a constant and often 1; for example, the discrete variable x may have the values x₀ = a, x₁ = a + 1, x₂ = a + 2, . . . , x_n = a + n. The function y has the corresponding values y₀, y₁, y₂, . . . , y_n, from which the differences can be found:

      Any equation that relates the values of Δy_i to each other or to x_i is a difference equation. In general, such an equation takes the form

      Systematic methods have been developed for the solution of these equations and for those in which, for example, second-order differences are involved. A second-order difference is defined as

 

* * *