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# Backward Difference operator(âˆ‡)

The operator âˆ‡ is called backward difference operator and pronounced as nepla.

Backward Difference operator ( âˆ‡) :

Let y = f(x) be a given function of x. Let y 0 , y1,..., yn be the values of y at

x= x0 , x1 , x2 ,..., xn respectively. Then

y1 âˆ’ y0 = âˆ‡y1

y 2 âˆ’ y1 = âˆ‡y2

y n âˆ’ ynâˆ’1 = âˆ‡yn

are called the first(backward) differences.

The operator âˆ‡ is called backward difference operator and pronounced as nepla.

Second(backward) differences: âˆ‡ 2 y n = âˆ‡ y n âˆ’ âˆ‡yn+1 , n = 1,2,3,â€¦

Third (backward) differences: âˆ‡ 3 y n = âˆ‡ 2 yn âˆ’ âˆ‡2 ynâˆ’1 n = 1,2,3,â€¦

In general, kth (backward) differences: âˆ‡ k yn = âˆ‡ k âˆ’1 yn âˆ’ âˆ‡kâˆ’1 ynâˆ’1 n = 1,2,3,â€¦

## Backward difference table: Backward differences can also be defined as follows.

âˆ‡ f (x) = f (x) âˆ’ f (x âˆ’ h)

First differences:  âˆ‡ f (x + h) = f (x + h) âˆ’ f (x)

âˆ‡ f (x + 2h) = f (x + 2h) âˆ’ f (x + h),...,h is the interval of spacing.

Second differences:

âˆ‡ 2 f (x + h) = âˆ‡(âˆ‡f (x + h) = âˆ‡( f (x + h) âˆ’ f (x))

= âˆ‡ f (x + h) âˆ’ âˆ‡f (x)

âˆ‡ 2 f (x + 2h)         = âˆ‡ f (x + 2h ) âˆ’ âˆ‡f (x + h)

Third differences:

âˆ‡ 3 f (x + h) = âˆ‡ 2 f (x + h ) âˆ’ âˆ‡2 f (x)

âˆ‡ 3 f (x + 2h) = âˆ‡ 2 f (x + 2h ) âˆ’ âˆ‡2 f (x + h)

Here we note that, âˆ‡ f (x + h) = f (x + h ) âˆ’ f (x) = Î”f (x)

âˆ‡ f (x + 2h) = f (x + 2h ) âˆ’ f (x + h) = Î”f (x + h)

âˆ‡ 2 f (x + 2h) = âˆ‡ f (x + 2h ) âˆ’ âˆ‡f (x + h) = Î”f (x + h ) âˆ’ Î”f (x)

= Î”2 f (x)

In general, âˆ‡n f (x + nh)= Î”n f (x)

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 : Backward Difference operator(âˆ‡) | Finite Differences | Numerical Methods