finite difference
Definition of .
The derivative of a function
is defined to be the expression
which makes sense whenever is differentiable (at least at ).However, the expression
makes sense even without being continuous, as long as .The expression is called a finite difference. The simplestcase when , written
is called the forward difference of . For other non-zero, we write
When , it is calleda backward difference of , sometimes written .Given a function and a real number , if we define and , then we have
Conversely, given and , we can find such that .
Some Properties of .
It is easy to see that the forward difference operator is linear:
- 1.
- 2.
, where is aconstant.
also has the properties
- 1.
for any real-valued constant function
, and
- 2.
for the identity function
.constant.
The behavior of in this respect is similar to that of thederivative operator. However, because the continuity of is not assumed, does not imply that is a constant. is merely a periodic function .Other interesting properties include
- 1.
for any real number
- 2.
where denotes the falling factorial
polynomial
- 3.
, where is the Bernoulli polynomial
of order .
From , we can also form other operators. For example, wecan iteratively define
(1) | |||
(2) |
Of course, all of the above can be readily generalized to .It is possible to show that can be written as a linear combination of
Difference Equation.
Suppose is a real-valued functionwhose domain is the -dimensional Euclidean space. Adifference equation (in one variable ) is the equation ofthe form
where is a one-dimensional real-valued function of .When are all integers, the expression on the left hand side ofthe difference equation can be re-written and simplified as
Difference equations are used in many problems in the real world,one example being in the study of traffic flow.
Title | finite difference |
Canonical name | FiniteDifference |
Date of creation | 2013-03-22 15:35:00 |
Last modified on | 2013-03-22 15:35:00 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 65Q05 |
Related topic | Equation |
Related topic | RecurrenceRelation |
Related topic | IndefiniteSum |
Related topic | DifferentialPropositionalCalculus |
Defines | forward difference |
Defines | backward difference |
Defines | difference equation |