# difference equation

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 *x _{n}*, we define Δ

*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}) - 2

*f*(

*x*

_{n+1}) +

*f*(

*x*); and so forth for Δ

_{n}*f*(

*x*), ..., Δ

_{n}*(*

^{m}f*x*).

_{n}

A difference table may be constructed showing values of Δ*f*(*x _{n}*),
Δ

^{2}

*f*(

*x*), ..., Δ

_{n}*(*

^{m}f*x*), ..., 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.