A difference equation is an equation that describes how something changes in discrete time steps. In the calculus of finite differences, a difference equation plays a role analogous to that played by a differential equation in calculus. The calculus of finite differences deals with discrete quantities; unlike calculus which deals with continuous quantities (see continuity).
For a function f(x) at a particular value xn, we define Δf(xn) as f(xn+1) - f(xn), where Δ is called the difference operator. From this, we find that Δ2f(xn) – i.e., Δ(f(xn+1) - f(xn) – can be expressed as f(xn+2) - 2f(xn+1) + f(xn); and so forth for Δf(xn), ..., Δmf(xn).
A difference table may be constructed showing values of Δf(xn), Δ2f(xn), ..., Δmf(xn), ..., and from this a relationship between the differences may be deduced.
Generally, then, a difference equation is any equation that expresses such a relationship; and use may be made of it to find discrete values for f(x) which lie outside the known range. Approximation of a differential equation to a suitable difference equation is often a powerful tool in the solution of the former.
Numerical solutions to integrals are usually realized as difference equations.