请输入您要查询的字词:

 

单词 FiniteDifference
释义

finite difference


Definition of Δ.

The derivativePlanetmathPlanetmath of a functionMathworldPlanetmath f: is defined to be the expression

dfdx:=limh0f(x+h)-f(x)h,

which makes sense whenever f is differentiableMathworldPlanetmathPlanetmath (at least at x).However, the expression

f(x+h)-f(x)h

makes sense even without f being continuousMathworldPlanetmathPlanetmath, as long as h0.The expression is called a finite difference. The simplestcase when h=1, written

Δf(x):=f(x+1)-f(x),

is called the forward differencePlanetmathPlanetmath of f. For other non-zeroh, we write

Δhf(x):=f(x+h)-f(x)h.

When h=-1, it is calleda backward difference of f, sometimes written f(x):=Δ-1f(x).Given a function f(x) and a real number h0, if we define y=xh and g(y)=f(hy)h, then we have

Δg(y)=Δhf(x).

Conversely, given g(y) and h0, we can find f(x) such that Δg(y)=Δhf(x).

Some Properties of Δ.

It is easy to see that the forward difference operator Δ is linear:

  1. 1.

    Δ(f+g)=Δ(f)+Δ(g)

  2. 2.

    Δ(cf)=cΔ(f), where c is aconstant.

Δ also has the properties

  1. 1.

    Δ(c)=0 for any real-valued constant functionMathworldPlanetmath c, and

  2. 2.

    Δ(I)=1 for the identity functionMathworldPlanetmath I(x)=x.constant.

The behavior of Δ in this respect is similarPlanetmathPlanetmath to that of thederivative operator. However, because the continuity of f is not assumed, Δf=0 does not imply that f is a constant. f is merely a periodic function f(x+1)=f(x).Other interesting properties include

  1. 1.

    Δax=(a-1)ax for any real number a

  2. 2.

    Δx(n)=nx(n-1) where x(n) denotes the falling factorialDlmfMathworld polynomialMathworldPlanetmathPlanetmathPlanetmath

  3. 3.

    Δbn(x)=nxn-1, where bn(x) is the Bernoulli polynomialDlmfDlmfPlanetmathPlanetmath of order n.

From Δ, we can also form other operators. For example, wecan iteratively define

Δ1f:=Δf(1)
Δkf:=Δ(Δk-1f),where k>1.(2)

Of course, all of the above can be readily generalized to Δh.It is possible to show that Δhf can be written as a linear combinationMathworldPlanetmath of

Δf,Δ2f,,Δhf.

Difference Equation.

Suppose F:n is a real-valued functionwhose domain is the n-dimensional Euclidean space. Adifference equation (in one variable x) is the equation ofthe form

F(x,Δh1k1f,Δh2k2f,,Δhnknf)=0,

where f:=f(x) is a one-dimensional real-valued function of x.When hi are all integers, the expression on the left hand side ofthe difference equation can be re-written and simplified as

G(x,f,Δf,Δ2f,,Δmf)=0.

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

Titlefinite difference
Canonical nameFiniteDifference
Date of creation2013-03-22 15:35:00
Last modified on2013-03-22 15:35:00
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id11
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 65Q05
Related topicEquation
Related topicRecurrenceRelation
Related topicIndefiniteSum
Related topicDifferentialPropositionalCalculus
Definesforward difference
Definesbackward difference
Definesdifference equation
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 17:31:10