A recurrence relation or a difference equation is the discrete analog to differential equations.
Definition: First Order Difference Equation
A first order difference equation is a recursively defined sequence in the form
What makes this first order is that we only need to know the most recent previous value to find the next value. It also has clear parallel to the first-order ODEs.
Definition: K-th Order Difference Equation
A difference equation of order
( ) has the form of
Examples
Example
Sometimes we use a different notation:
Example: Pascal's rule
Linear Difference Equation
In a linear difference equation, the
with initial conditions
Modeling Discrete Systems
Difference equations are great ways to model discrete systems as they are mathematically concise and precise, and they are also convenient for step-by-step analysis. However, deriving a closed-form solution from difference equations is often challenging, and sometimes a more imperative description like the block diagram may be more preferable to describe a system.