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^{-1}*f *(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

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12th Business Maths and Statistics : Chapter 5 : Numerical Methods : Relations between the operators Î”, âˆ‡ and E | Finite Differences | Numerical Methods

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