Definition:Finite Difference Operator/Forward Difference/Second

Definition

Let $f: \R \to \R$ be a real function.

Let $y = \map f x$ have known values:

$y_k = \map f {x_k}$

for $x_k \in \set {x_0, x_1, \ldots, x_n}$ defined as:

$x_k = x_0 + k h$

for some $h \in \R_{>0}$.

$\ds \map {\Delta^2 f} {x_i}$

$=$

$\ds \map \Delta {\map {\Delta f} {x_i} }$

$\ds $

$=$

$\ds \Delta \map f {x_{i + 1} } - \Delta \map f {x_i}$

$\ds $

$=$

$\ds \map f {x_{i + 2} } - 2 \Delta \map f {x_{i + 1} } + \Delta \map f {x_i}$

for $i = 0, 1, 2, \ldots, n - 2$