Finite difference

In mathematics, a finite difference is like a differential quotient, except that it uses finite quantities instead of infinitesimal ones.

Numerical analysis

Another important aspect is that finite differences approach differential quotients as h goes to zero. Thus, we can use finite differences to approximate derivatives. This is often used in numerical analysis, especially in numerical ordinary differential equations and numerical partial differential equations, which aim at the numerical solution of ordinary and partial differential equations respectively. The resulting methods are called finite-difference methods.

Related Topics:
Numerical analysis - Numerical ordinary differential equations - Numerical partial differential equations - Ordinary - Partial differential equation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For example, consider the ordinary differential equation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: u'(x) = 3u(x) + 2. ,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The Euler method for solving this equation uses the finite difference

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:rac{u(x+h) - u(x)}{h} pprox u'(x)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

to approximate the differential equation by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: u(x+h) = u(x) + h(3u(x)+2). ,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The last equation is called a finite-difference equation. Solving this equation gives an approximate solution to the differential equation.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The error between the approximate solution and the true solution is determined by the error that is made by going from a differential operator to a difference operator. This error is called the discretization error or truncation error (the term truncation error reflects the fact that a difference operator can be viewed as a finite part of the infinite Taylor series of the differential operator).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~