Question

In the difference equation the operator E is defined
as​

Answer 1

Difference Operators

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, . . . xn as f0, f1, . . . fn 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) orDfi = fi+1 - fi

where any typical fi = f(xi).

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)

= fi+2 - 2fi+1 + fi

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