## difference equationAn 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 x, we define Δ_{n}f(x)
as _{n}f(x_{n+1}) - f(x),
where Δ is called the difference operator. From this, we find that
Δ_{n}^{2}f(x) – i.e., Δ(_{n}f(x_{n+1})
- f(x) – can be expressed as _{n}f(x_{n+2})
- 2f(x_{n+1}) + f(x);
and so forth for Δ_{n}f(x), ..., Δ_{n}(^{m}fx).
_{n}A difference table may be constructed showing values of Δ f(x),
Δ_{n}^{2}f(x), ..., Δ_{n}(^{m}fx),
..., and from this a relationship between the differences may be deduced.
_{n}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. ## Related categories• APPROXIMATIONS AND AVERAGES• CALCULUS AND ANALYSIS | |||||

Home • About • Copyright © The Worlds of David Darling • Encyclopedia of Alternative Energy • Contact |