The symbol Î” is called the forward difference operator and pronounced as delta.

**Forward Difference Operator(âˆ† ):**

Let *y* = *f*(*x*) be a given function of *x*.
Let *y* _{0} , *y*_{1} , *y* _{2} , â€¦.. ,
*y _{n}* be the values of

(i.e) Î”*y* _{0} =
*y*_{1} âˆ’
*y* _{0} , Î”*y*_{1} = *y* _{2} âˆ’ *y*_{1}, Î”*y* _{2} = *y*_{3} âˆ’ *y* _{2}
,..., Î”*y _{n}*

In general, Î”*y* * _{n}* =

The symbol Î”
is called the forward difference operator and pronounced as delta.

The forward difference operator âˆ† can also be defined as D*f* ( *x*) = *f* ( *x* + *h* ) âˆ’ *f* ( *x*), *h* is the equal
interval of spacing.

**Properties of the operator **Î”** :**

**Property 1: **If** ***c*** **is a constant then** **Î”*c*** **=** **0

**Proof: **Let *f* (*x*) =
*c*

âˆ´* f *(* x *+* h *)* *=* c *(where â€˜*h*â€™ is the
interval of difference)

Î”*f *(* x*)* *=* f *(* x *+* h *)* *âˆ’* f *(* x*)

Î”*c* = *c* âˆ’ *c* = 0

**Property 2: **Î” is distributive i.e. Î” ( *f* (*x*) + *g* (*x*)) = Î”* f* (*x*) + Î”* g* (*x*)

**Proof: Î”[***f *(*x*)* *+* g *(*x*)] = [*f *(*x *+* h *)* *+* g*(*x *+* h*) ] â€“ [*f *(*x*)+* g *(*x*)]

= *f *(*x *+* h *)* *+* g*(*x *+* h*)* *âˆ’* f *(*x*)* *âˆ’* g *(*x*)

*= f *(*x *+* h *)* *âˆ’* f *(*x *)+* g*(*x *+* h*)* *âˆ’* g *(*x*)

=** Î”*** f *(*x*)* *+* ***Î”*** g *(*x*)

Similarly we can show that **Î”*** *[*f *(* x*)* *âˆ’* g *(* x*)]* *= *f *(*x*)* *âˆ’* ***Î”*** *[*g *(* x*)]

In general,** ****Î”[** *f*_{1}* *(*x*)* *+* f *_{2}* *(*x*)......* *+ *f _{n} *(

**Property 3: **If** ***c*** **is a constant then** **Î”** ***c f*** **(** ***x*** **)** **=** ***c*** **Î”** ***f*** **(** ***x*)

**Proof: Î”** [*c f* (*x*)]
= *c f* ( *x* + *h*) âˆ’ *c f* ( *x*)

= *c* [ *f* (*x* + *h*) âˆ’
*f* (*x*)]

= *c* **Î”*** f* (*x*)

1.
If *m* and *n* are positive integers then **Î”*** ^{ m}*
.

2.
**Î”[*** f *(*x*) *g *(*x*) ] = *f* (*x*) *g* (*x*) + *g* (*x*) **Î”*** f* (*x*)

3.

The differences of the first differences denoted by Î”^{2}y_{0},
Î”^{2}y_{1}, â€¦., Î”^{2}y_{n}, are called second differences, where

Similarly the differences of second differences are called third differences.

It is convenient to represent the above differences in a table as
shown below.

Tags : Finite Differences | Numerical Methods , 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

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12th Business Maths and Statistics : Chapter 5 : Numerical Methods : Forward Difference Operator(âˆ†) | Finite Differences | Numerical Methods

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