Page 41 -

2

Difference Equations

This chapter introduces difference equations and examines some simple but
important cases of their applications. We develop simple algorithms for their
numerical solutions and apply these techniques to the solution of some prob-
lems of interest to the engineering professional. In particular, it illustrates
each type of difference equation that is of widespread interest.

2.1 Simple Linear Forms

The following components are needed to define and solve a difference
equation:

1. An ordered array defining an index for the sequence of elements
2. An equation connecting the value of an element having a certain
index with the values of some of the elements having lower indices
(the order of the equation being defined by the number of lower
indices terms appearing in the difference equation)
3. A sufficient number of the values of the elements at the lowest
indices to act as seeds in the recursive generation of the higher
indexed elements.

For example, the Fibonacci numbers are defined as follows:

1. The ordered array is the set of positive integers
2. The defining difference equation is of second order and is given by:

F(k + 2) = F(k + 1) + F(k) (2.1)

3. The initial conditions are F(1) = F(2) = 1 (note that the required
number of initial conditions should be the same as the order of the
equation).

0-8493-????-?/00/$0.00+$.50
© 2000 by CRC Press LLC
© 2001 by CRC Press LLC