What is the difference between Difference equations and Recurrence relations?

Question

Is there any difference between Difference equations and Recurrence relations?
Some people are use them as difference equations and some are use as recurrence relations.
I couldn't find in anywhere. Also I am looking for good reference for first order first degree nonlineardifference equations/Recurrence relations. Thank you.

Answer 1

There is indeed a difference between difference equations and recurrence relations. In fact, those methods are both equivalent and many books choose to call a relation either a difference one or a recurrence one.

As you may know, a recurrence relation is a relation between terms of a sequence. For example : $ 4*x_{n} = 3*x_{n-1} + 2$. Given a certain number belonging to the sequence, you may find the following one.

A difference equation is defined as :

$ \Delta(a_{n}) = a_{n+1} - a_{n} $ for k = 1

$ \Delta^{2}(a_{n}) = \Delta(a_{n+1}) - \Delta(a_{n}) $ for k = 2

And more generally, $ \Delta^{k}(a_{n}) = \Delta^{k-1}(a_{n+1}) - \Delta^{k-1}(a_{n}) $

So you can also define a series with your difference equations. Most of the time, you can define a series both ways. Just choose the 'syntax' you want to define it. If you choose one, let's say recurrence, you'll be able to switch to a difference equation and vice versa.

Answer 2

A recurrence relation is a difference equation in explicit form that describes one element of the series in terms of other smaller ones and are usually related to recursion. Then only the first example would be a recurrence relation:

$x_{n+1}=0.5x_n+1$

$x_{n+1}-0.5x_n=1$