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Finite Differences | Numerical Methods - Backward Difference operator(∇) | 12th Business Maths and Statistics : Chapter 5 : Numerical Methods

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

Backward Difference operator(∇)

The operator ∇ is called backward difference operator and pronounced as nepla.

Backward Difference operator ( ) :

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 y0 = y1

y 2 y1 = y2

y n yn1 = yn

are called the first(backward) differences.

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

Second(backward) differences: 2 y n = y n yn+1 , n = 1,2,3,…

Third (backward) differences: 3 y n = 2 yn 2 yn1 n = 1,2,3,…

In general, kth (backward) differences: k yn = k 1 yn k1 yn1 n = 1,2,3,…

Backward difference table:


Backward differences can also be defined as follows.

f (x) = f (x) f (x h)

First differences:  f (x + h) = f (x + h)f (x)

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

Second differences:

2 f (x + h) = (f (x + h) = ( f (x + h) f (x))

= ∇ f (x + h) f (x)

2 f (x + 2h)         = f (x + 2h ) f (x + h)

Third differences:

3 f (x + h) = 2 f (x + h ) 2 f (x)

3 f (x + 2h) = 2 f (x + 2h ) 2 f (x + h)

Here we note that, f (x + h) = f (x + h ) f (x) = Δf (x)

f (x + 2h) = f (x + 2h ) f (x + h) = Δf (x + h)

2 f (x + 2h) = f (x + 2h ) f (x + h) = Δf (x + h ) Δf (x)

= Δ2 f (x)

In general, n f (x + nh)= Δn f (x)

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