Home | | Business Maths 12th Std | Relations between the operators Î”, âˆ‡ and E

# Relations between the operators Î”, âˆ‡ and E

Relations between the operators : Forward Difference Operator(âˆ†), Backward Difference operator(âˆ‡), Shifting operator (E)

Relations between the operators Î”, âˆ‡ and E:

1. Î” â‰¡ E âˆ’1

Proof:         From the definition of Î” we know that

Î” f (x) = f (x + h ) âˆ’ f (x) and

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

where h is the interval of difference.

Î” f (x) = f (x + h ) âˆ’ f (x)

Î” f (x) = Ef (x ) âˆ’ f (x)

â‡’ Î” f (x) = (E âˆ’1) f (x)

Î” â‰¡ E âˆ’1

âˆ´    E â‰¡ 1 + Î”

2.   Î” E       â‰¡       Î” E

Proof:

E(Î”f(x)) = E[f(x+h)- f(x)]

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

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

=Î”f(x+h)

=Î”Ef(x)

Î”E â‰¡ Î”E

3. âˆ‡â‰¡ E âˆ’1 / E

âˆ‡f (x) = f (x) - f (x-h)

= f (x) â€“ E-1f (x)

= (1- E-1) f (x)

âˆ‡ â‰¡ (1- E-1)

âˆ‡ â‰¡ 1 â€“ 1/E

Hence âˆ‡ â‰¡ [E â€“ 1]/E Tags : Finite Differences | Numerical Methods , 12th Business Maths and Statistics : Chapter 5 : Numerical Methods
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Business Maths and Statistics : Chapter 5 : Numerical Methods : Relations between the operators Î”, âˆ‡ and E | Finite Differences | Numerical Methods