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 x_{0} , x_{0}+h , x_{0}+2h , x_{0}+3h
,â€¦., x_{0}+nh be the consecutive values of x. Then the operator E is
defined as

E[*f*(x_{0})] =
f(x_{0}+h)

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

*E*[* f *(* x *_{0 }+* h*)] = *f* (*x* _{0}
+ 2*h*), *E*[ *f*
(*x*_{0} +
2*h*)] = *f* (*x*_{0} + 3*h*),...,

E[ *f* (*x*_{0} + (*n* âˆ’1)*h*)]
= *f* (*x*_{0} +
*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) E^{2}*f *(*x*) = E[*E f *(*x*)]
= E [ *f* (*x* +
*h* ) = *f* (*x* + 2*h*)

In general ,

E^{n}*f* (*x*) = *f* (*x* + *nh*) and E^{âˆ’}^{n}*f* (*x*) = *f* (*x* âˆ’ *nh*)

1. E[*f*_{1}(*x*)+*
f*_{2}(*x*)+â€¦.* F*_{n}(*x*)] = E* f*_{1}(*x*) + E* f*_{2}(*x*)â€¦+â€¦..+E[* f*_{n}(*x*)]

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

3. E * ^{m}* [ E

4. If â€˜*n*â€™ is a positive integer, then E* ^{n}* [ E

**Note**

Let y = *f*(*x*) be given function of
x. Let y_{0}, y_{1}, y_{2},â€¦. y_{n} be the
values of y at x = x_{0}, x_{1},
x_{2},â€¦.. x_{n}.. Then E can also be defined as

Ey_{0} = y_{1}, Ey_{1}=y_{2},â€¦.,Ey_{n-1}=y_{n}

E[ Ey_{0}] = E(y_{1})
= y_{2} and in general E^{n}y_{0=}y_{n}

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 : Shifting operator (E) | Finite Differences | Numerical Methods

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