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# finite difference

Definition of $\Delta$.

The derivative of a function $f\colon\mathbb{R}\to\mathbb{R}$ is defined to be the expression

$\frac{df}{dx}:=\lim_{{h\to 0}}\frac{f(x+h)-f(x)}{h},$ |

which makes sense whenever $f$ is differentiable (at least at $x$). However, the expression

$\frac{f(x+h)-f(x)}{h}$ |

makes sense even without $f$ being continuous, as long as $h\neq 0$.
The expression is called a *finite difference*. The simplest
case when $h=1$, written

$\Delta f(x):=f(x+1)-f(x),$ |

is called the *forward difference* of $f$. For other non-zero
$h$, we write

$\Delta_{h}f(x):=\frac{f(x+h)-f(x)}{h}.$ |

When $h=-1$, it is called
a *backward difference* of $f$, sometimes written $\nabla f(x):=\Delta_{{-1}}f(x)$.
Given a function $f(x)$ and a real number $h\neq 0$, if we define $y=\frac{x}{h}$ and $g(y)=\frac{f(hy)}{h}$, then we have

$\Delta g(y)=\Delta_{h}f(x).$ |

Conversely, given $g(y)$ and $h\neq 0$, we can find $f(x)$ such that $\Delta g(y)=\Delta_{h}f(x)$.

Some Properties of $\Delta$.

It is easy to see that the forward difference operator $\Delta$ is linear:

1. $\Delta(f+g)=\Delta(f)+\Delta(g)$

2. $\Delta(cf)=c\Delta(f)$, where $c\in\mathbb{R}$ is a constant.

$\Delta$ also has the properties

1. $\Delta(c)=0$ for any real-valued constant function $c$, and

2. $\Delta(I)=1$ for the identity function $I(x)=x$. constant.

The behavior of $\Delta$ in this respect is similar to that of the derivative operator. However, because the continuity of $f$ is not assumed, $\Delta f=0$ does not imply that $f$ is a constant. $f$ is merely a periodic function $f(x+1)=f(x)$. Other interesting properties include

1. $\Delta a^{x}=(a-1)a^{x}$ for any real number $a$

2. $\Delta x^{{(n)}}=nx^{{(n-1)}}$ where $x^{{(n)}}$ denotes the falling factorial polynomial

3. $\Delta b_{n}(x)=nx^{{n-1}}$, where $b_{n}(x)$ is the Bernoulli polynomial of order $n$.

From $\Delta$, we can also form other operators. For example, we can iteratively define

$\displaystyle\Delta^{{1}}f:=\Delta f$ | (1) | ||

$\displaystyle\Delta^{{k}}f:=\Delta(\Delta^{{k-1}}f),\quad\mbox{where }k>1.$ | (2) |

Of course, all of the above can be readily generalized to $\Delta_{h}$. It is possible to show that $\Delta_{h}f$ can be written as a linear combination of

$\Delta f,\Delta^{2}f,\ldots,\Delta^{h}f.$ |

Difference Equation.

Suppose $F\colon\mathbb{R}^{n}\to\mathbb{R}$ is a real-valued function
whose domain is the $n$-dimensional Euclidean space. A
*difference equation* (in one variable $x$) is the equation of
the form

$F(x,\Delta_{{h_{1}}}^{{k_{1}}}f,\Delta_{{h_{2}}}^{{k_{2}}}f,\ldots,\Delta_{{h_% {n}}}^{{k_{n}}}f)=0,$ |

where $f:=f(x)$ is a one-dimensional real-valued function of $x$. When $h_{i}$ are all integers, the expression on the left hand side of the difference equation can be re-written and simplified as

$G(x,f,\Delta f,\Delta^{{2}}f,\ldots,\Delta^{{m}}f)=0.$ |

Difference equations are used in many problems in the real world, one example being in the study of traffic flow.

## Mathematics Subject Classification

65Q05*no label found*

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