PlanetMath: finite difference

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finite difference

(Definition)

Definition of $\Delta$ .

The derivative of a function $f\colon\mathbb{R}\to\mathbb{R}$ is defined to be the expression

$\displaystyle \frac{df}{dx}:=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},$

which makes sense whenever

is differentiable (at least at

). However, the expression

$\displaystyle \frac{f(x+h)-f(x)}{h}$

makes sense even without

being continuous, as long as $h\neq 0$ . The expression is called a finite difference. The simplest case when

, written

$\displaystyle \Delta f(x):=f(x+1)-f(x),$

is called the forward difference of

. For other non-zero

, we write

$\displaystyle \Delta_h f(x):=\frac{f(x+h)-f(x)}{h}.$

When

, it is called a backward difference of

, sometimes written $\nabla f(x):=\Delta_{-1} f(x)$ . Given a function

and a real number $h\neq 0$ , if we define $y=\frac{x}{h}$ and $g(y)=\frac{f(hy)}{h}$ , then we have

$\displaystyle \Delta g(y)=\Delta_h f(x).$

Conversely, given

and $h\neq 0$ , we can find

such that $\Delta g(y)=\Delta_h f(x)$ .

Some Properties of $\Delta$ .

It is easy to see that the forward difference operator $\Delta$ is linear:

$\Delta(f+g)=\Delta(f)+\Delta(g)$
$\Delta(cf)=c\Delta(f)$ , where $c\in\mathbb{R}$ is a constant.

$\Delta$ also has the properties

$\Delta(c)=0$ for any real-valued constant function , and
$\Delta(I)=1$ for the identity function . constant.

The behavior of $\Delta$ in this respect is similar to that of the derivative operator. However, because the continuity of

is not assumed, $\Delta f=0$ does not imply that

is a constant.

is merely a periodic function

. Other interesting properties include

$\Delta a^x=(a-1)a^x$ for any real number
$\Delta x^{(n)}=nx^{(n-1)}$ where $x^{(n)}$ denotes the falling factorial polynomial
$\Delta b_n(x)=nx^{n-1}$ , where is the Bernoulli polynomial of order .

From $\Delta$ , we can also form other operators. For example, we can iteratively define

		$\displaystyle \Delta^{1}f:=\Delta f$	(1)
		$\displaystyle \Delta^{k}f:=\Delta(\Delta^{k-1}f),$ where $\displaystyle k>1.$	(2)

Of course, all of the above can be readily generalized to $\Delta_h$ . It is possible to show that $\Delta_h f$ can be written as a linear combination of

$\displaystyle \Delta f,\Delta^2 f,\ldots,\Delta^h f.$

Difference Equation.

Suppose $F\colon\mathbb{R}^n\to\mathbb{R}$ is a real-valued function whose domain is the -dimensional Euclidean space. A difference equation (in one variable ) is the equation of the form

$\displaystyle F(x,\Delta_{h_1}^{k_1}f,\Delta_{h_2}^{k_2}f,\ldots,\Delta_{h_n}^{k_n}f)=0,$

where

is a one-dimensional real-valued function of

. When

are all integers, the expression on the left hand side of the difference equation can be re-written and simplified as

$\displaystyle G(x,f,\Delta f,\Delta^{2}f,\ldots,\Delta^{m}f)=0.$

Difference equations are used in many problems in the real world, one example being in the study of traffic flow.

"finite difference" is owned by CWoo.

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Also defines:

forward difference, backward difference, difference equation

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Cross-references: flow, left hand side, integers, equation, variable, Euclidean space, domain, linear combination, order, Bernoulli polynomial, polynomial, falling factorial, periodic function, imply, similar, identity function, constant function, operator, easy to see, properties, real number, continuous, even, differentiable, expression, function, derivative
There are 4 references to this entry.

This is version 8 of finite difference, born on 2005-11-18, modified 2005-12-13.
Object id is 7493, canonical name is FiniteDifference.
Accessed 5785 times total.

Classification:

AMS MSC:

65Q05 (Numerical analysis :: Difference and functional equations, recurrence relations)

Pending Errata and Addenda

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