Question

define operator E,∆ and ∇​

Answer 1

There are several varieties of finite difference operators, the most common of which are forward difference, backward difference, and central difference operators.

• The shift operator is denoted by E and is given by
  $Ef(x) = f(x + h)$
  In terms of y, the above formula becomes the following
  $Ey_i = y_i_+_1$
  Note that shift operator increases subscript of y by one and when it is applied twice on the function f(x), then the subscript of y is increased by 2 i.e. E²f(x) = E[Ef(x)] = E[f(x + h)] = f(x + 2h)
  In general,$E^nf(x) = f(x + nh)$ or $E^ny_i = y_i_+_n_h.$

• The sign ∆  is known as the forward difference operator and is pronounced delta. The forward difference operator is sometimes defined as Df(x) = f (x + h) f (x), where h is the equal spacing interval.

• Backward differences or ∇ are defined as  the interpolation polynomial of order n through the points $y_0, y_-_1, y_-_2$,... is. The value $a=0$ gives $x= x_0$ $a=1$ gives $x=x_1$ and this approximation uses the points to the left of the point $x_0$, and that fits a polynomial through the two or more than two points.

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