We have already seen one difference operator called divided difference operator in the earlier section. We define few more difference operators and their properties in this section.

Forward difference operator :

Suppose that a fucntion f(x) is given at equally spaced discrete points say x₀, x₁, . . . x_n as f₀, f₁, . . . f_n respectively. Also let the constant difference between two consecutive points of x is called the interval of differencing or the step length denoted by h. Then the forward difference operator D is defined as 

Df(x) = f(x + h) - f(x) orDf_i = f_i+1 - f_i

where any typical f_i = f(x_i).

Similarly the higher differences are defined as

D²f(x)    =    D(Df(x))
               =    D(f(x + h) - f(x))
               =    f(x + 2h) - 2f(x + h) + f(x)
               =    f_i+2 - 2f_i+1 + f_i

or in general,                         Dⁿ f(x) = D^n-1(Df(x)) = D^n-1f(x + h) - D^n-1f(x)

Example :

Let a function f(x) is given at the points (0, 7), (4, 43), (8,367) then find the forward difference of the function at x = 4. 

Df(x) = f(x + h) - f(x)

ÞDf(4) = f(4 + h) - f(4)
              = f(8) - f(4) = 367 - 43 = 324 

Example :

Find Df(x) for the function x² + 2x + 3 with h = 2

Df(x)  = f(x + h) - f(x)    =    (x + 2)² + 2(x + 2) + 3 - x² - 2x - 3
                                       =   x² + 4x + 4 +  2x  + 4 +  3  -  x²  -  2x  -  3     =   4x + 8

D²f(x)  = 4(x + 2) + 8 - 4x - 8    = 4x + 8 - 4x = 8

D³f(x)  = 8 - 8 = 0

Properties :

1. D(f(x) + g(x))  = Df(x)  + Dg(x)
2. D(c f(x))  = c.Df(x)  where c is a constant
3. If f(x) is a polynomial of degree n, 

	n
f(x) =	S	ai xi	then  Dnf(x) is constant, and is equal to ann!hn
	i=0

Proof :

Lets prove this by mathematical induction 
for n = 1,    f(x) = a₀ + a₁x
Df(x) = a₀ + a₁(x + h) - a₀ - a₁x = a₁h
          = a₁1!h¹
assume that the formula is true for all polynomials upto the degree n-1. Consider the n^th degree polynomial 

            = 0 + a_n Dⁿ xⁿ
            = 0 + a_n D^n-1((x + h)ⁿ - xⁿ)
            = 0 + a_n D^n-1( nhx^n-1 - g(x))

where g(x) is a polynomial of degree less than (n-1) and hence D^n-1g(x) = 0
ÞDⁿ f(x) = a_n D^n-1( nhx^n-1)
                = a_n(nh)(n-1)!h^n-1
                = a_nn!hⁿ. 

Backward difference operator : Ñ

The difference operator Ñ defined by 

Ñf(x) = f(x) - f(x-h)

Ñf_i = f_i - f_i-1

is called backward difference operator. 

Ѳf_i = Ñ(Ñf_i)  = Ñ(f_i - f_i-1)

= f_i - 2f_i-1 + f_i-2 and so on.

Central difference operator : d

The difference operator d  defined by 

df(x) = f(x + h/2) - f(x - h/2)

df_i = f_{i + 1/2} - f_{i - 1/2}

is called the central difference operator. 

d²f_i = d(f_{i + 1/2} - f_{i - 1/2})  = (f_i+1 - f_i) - (f_i - f_i-1)

        = f_i+1 - 2f_i + f_i-1

Shift operator :

Averaging operator :

Some relations between various operators :

1. E = 1 + D 

 

 
 
 

Ef_i = f_i+1 =  f_i+1 - f_i + f_i = (1 + D)f_i

Þ E = 1 + D
 

	¥
Enfi =	S	(	n i	)	Di fi
	k=0

for n = 1,  E¹f_i = f_i+1 = (1 + D)f_i
 

	¥
RHS is	S	(	1 i	)	Di fi	= D0 fi+D1 fi = (1+D)fi
	k=0

Hence the result is true for n = 1, Assume now that the result is true for the value (n-1) then 
 
 

	¥
En-1fi =	S	(	n-1 i	)	Di fi
	k=0

consider 

Eⁿf_i = E E^n-1f_i
 

	¥
= (1 + D)	S	(	n-1 i	)	Di fi
	k=0

¥         =  S ( n-1 i ) Di fi k=0

+

¥ S ( n-1 i ) Di+1 fi k=0

¥         =  S ( n-1 i ) Di fi k=0

+

¥ S ( n-1 j-1 ) Dj fi j=1

¥

=

S

[

(

n-1 i

)

+

(

n-1 i

)

]

Di fi

k=0

...

(

n-1 i

) = 0

and j is arbitrary

	¥
=	S	(	n i	)	Di fi
	k=0


 

	¥
ÞEnfi =	S	(	n i	)	Di fi
	k=0