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Finite Differences | Numerical Methods - Shifting operator (E) | 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

Chapter: 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

Shifting operator (E)

E is called the shifting operator . It is also called the displacement operator.

Shifting operator (E):

Let y = f (x) be a given function of x and x0 , x0+h , x0+2h , x0+3h ,…., x0+nh be the consecutive values of x. Then the operator E is defined as    

E[f(x0)] = f(x0+h)

E is called the shifting operator . It is also called the displacement operator.

E[ f ( x 0 + h)] = f (x 0 + 2h), E[ f (x0 + 2h)] = f (x0 + 3h),...,

E[ f (x0 + (n 1)h)] = f (x0 + nh)

E[ f (x)] = f (x + h) , h is the (equal) interval of spacing

E 2 f (x) means that the operator E is applied twice on f (x)

(i.e) E2f (x) = E[E f (x)] = E [ f (x + h ) = f (x + 2h)

In general ,

En f (x) = f (x + nh) and En f (x) = f (xnh)

 

Properties of the operator E:

1. E[f1(x)+ f2(x)+…. Fn(x)] = E f1(x) + E f2(x)…+…..+E[ fn(x)]

2. E[cf (x)] =cE[f(x)] c is constant

3. E m [ En f (x) ] = En [ ( Em f (x) ) = Em +n f ( x)

4. If ‘n’ is a positive integer, then En [ En ( f (x)) ] = f (x)


Note

Let y = f(x) be given function of x. Let y0, y1, y2,…. yn be the values of y at  x = x0, x1, x2,….. xn.. Then E can also be defined as

Ey0 = y1, Ey1=y2,….,Eyn-1=yn

E[ Ey0] = E(y1) = y2 and in general Eny0=yn

 

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