finite difference

finite difference

Definition of

Δ

.

The

derivative

of a

function

f

:

ℝ

→

ℝ

is defined to be the expression

d

⁢

f

d

⁢

x

:=

lim

h

→

0

⁡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

,

which makes sense whenever

f

is

differentiable

(at least at

x

).
However, the expression

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

makes sense even without

f

being

continuous

, as long as

h

≠

0

.
The expression is called a

finite difference

. The simplest
case when

h

=

1

, written

Δ

⁢

f

⁢

(

x

)

:=

f

⁢

(

x

+

1

)

-

f

⁢

(

x

)

,

is called the

forward

difference

of

f

. For other non-zero

h

, we write

Δ

h

⁢

f

⁢

(

x

)

:=

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

.

When

h

=

-

1

, it is called
a

backward difference

of

f

, sometimes written

∇

⁡

f

⁢

(

x

)

:=

Δ

-

1

⁢

f

⁢

(

x

)

.
Given a function

f

⁢

(

x

)

and a

real number

h

≠

0

, if we define

y

=

x

h

and

g

⁢

(

y

)

=

f

⁢

(

h

⁢

y

)

h

, then we have

Δ

⁢

g

⁢

(

y

)

=

Δ

h

⁢

f

⁢

(

x

)

.

Conversely

, given

g

⁢

(

y

)

and

h

≠

0

, we can find

f

⁢

(

x

)

such that

Δ

⁢

g

⁢

(

y

)

=

Δ

h

⁢

f

⁢

(

x

)

.

Some Properties of

Δ

.

It is easy to see that the forward difference operator

Δ

is linear:

1.

Δ

⁢

(

f

+

g

)

=

Δ

⁢

(

f

)

+

Δ

⁢

(

g

)

2.

Δ

⁢

(

c

⁢

f

)

=

c

⁢

Δ

⁢

(

f

)

, where

c

∈

ℝ

is a

constant

.

Δ

also has the properties

1.

Δ

⁢

(

c

)

=

0

for any real-valued

constant function

c

, and

2.

Δ

⁢

(

I

)

=

1

for the

identity function

I

⁢

(

x

)

=

x

.
constant.

The behavior of

Δ

in this respect is

similar

to that of the
derivative

operator

. However, because the

continuity

of

f

is not assumed,

Δ

⁢

f

=

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.

Δ

⁢

a

x

=

(

a

-

1

)

⁢

a

x

for any real number

a

2.

Δ

⁢

x

(

n

)

=

n

⁢

x

(

n

-

1

)

where

x

(

n

)

denotes the

falling factorial

polynomial

3.

Δ

⁢

b

n

⁢

(

x

)

=

n

⁢

x

n

-

1

, where

b

n

⁢

(

x

)

is the

Bernoulli polynomial

of order

n

.

From

Δ

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

Δ

1

⁢

f

:=

Δ

⁢

f

(1)

Δ

k

⁢

f

:=

Δ

⁢

(

Δ

k

-

1

⁢

f

)

,

where

⁢

k

>

1

.

(2)

Of course, all of the above can be readily generalized to

Δ

h

.
It is possible to show that

Δ

h

⁢

f

can be written as a

linear combination

of

Δ

⁢

f

,

Δ

2

⁢

f

,

…

,

Δ

h

⁢

f

.

Difference Equation

.

Suppose

F

:

ℝ

n

→

ℝ

is a real-valued function
whose domain is the

n

-dimensional

Euclidean space

. A

difference equation

(in one

variable

x

) is the equation of
the form

F

⁢

(

x

,

Δ

h

1

k

1

⁢

f

,

Δ

h

2

k

2

⁢

f

,

…

,

Δ

h

n

k

n

⁢

f

)

=

0

,

where

f

:=

f

⁢

(

x

)

is a one-dimensional real-valued function of

x

.
When

h

i

are all

integers

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

G

⁢

(

x

,

f

,

Δ

⁢

f

,

Δ

2

⁢

f

,

…

,

Δ

m

⁢

f

)

=

0

.

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

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