Definition:Finite Difference Operator/Forward Difference

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Definition

Finite-difference-preamble.png


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


First Forward Difference Operator

The first forward difference operator on $f$ is defined as:

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

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


Second Forward Difference Operator

The second forward difference operator on $f$ is defined as:

\(\ds \map {\Delta^2 f} {x_i}\) \(=\) \(\ds \map \Delta {\map {\Delta f} {x_i} }\)
\(\ds \) \(=\) \(\ds \Delta \map f {x_{i + 1} } - \Delta \map f {x_i}\)
\(\ds \) \(=\) \(\ds \map f {x_{i + 2} } - 2 \Delta \map f {x_{i + 1} } + \Delta \map f {x_i}\)

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


$k$th Forward Difference Operator

The $k$th forward difference operator on $f$ is defined as:

\(\ds \map {\Delta^k f} {x_i}\) \(=\) \(\ds \map \Delta {\map {\Delta^{k - 1} f} {x_i} }\)
\(\ds \) \(=\) \(\ds \Delta^{k - 1} \map f {x_{i + 1} } - \Delta^{k - 1} \map f {x_i}\)
\(\ds \) \(=\) \(\ds \sum_{s \mathop = 0}^k \paren {-1}^{k - s} \dbinom k s y_{i + s}\)

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


Examples

Value of $y_1$

The value of $y_1$ is given by:

$y_1 = y_0 + \Delta y_0$

where $\Delta y_0$ is the first forward difference of $y_0$.


Value of $y_2$

The value of $y_2$ is given by:

$y_2 = y_0 + 2 \Delta y_0 + \Delta^2 y_0$

where:

$\Delta y_0$ is the first forward difference of $y_0$
$\Delta^2 y_0$ is the second forward difference of $y_0$.


Value of $y_k$

The value of $y_k$ is given by:

$y_k = y_0 + \ds \sum_{s \mathop = 1}^k \dbinom k s \Delta^k y_0$

where $\Delta^k y_0$ is the $k$th forward difference of $y_0$.


Also known as

Some sources do not acknowledge the existence of the backward difference operator or central difference operator.

Such sources merely refer to the forward difference operator as the (finite) difference operator.


Also see

  • Results about the forward difference operator can be found here.


Sources

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