replaced by differences (discrete change), then linear ordinary differential equations
become linear difference equations. Let us begin our study of discrete phenomena
by investigating difference equations and determining ways to construct solutions to
such equations.
In the following discussions, note that the various techniques developed for an-
alyzing discrete systems are very similar to many of the methods used for studying
continuous systems.
Differences and Difference Equations
Consider the function y=f(x)illustrated in the figure 10-8 which is evaluated
at the equally spaced x−values of x 0 , x 1 , x 2 ,... , x i,... , x n, x n+1,... where xi+1 =xi+h
for i= 0 , 1 , 2 ,.. ., n where his the distance between two consecutive points.
Let yn=f(xn)and consider the approximation of the derivative dy
dxat the discretevalue xn. Use the definition of a derivative and write the approximation as
dy
dx∣∣
∣x=x
n≈yn+1 −yn
h.This is called a forward difference approximation. By letting h = 1 in the above
equation one can define the first forward difference of ynas
∆yn=yn+1 −yn. (10 .50)There is no loss in generality in letting h= 1 , as one can always rescale the x-axis by
defining the new variable Xdefined by the transformation equation x=x 0 +Xh , then
when x=x 0 , x 0 +h, x 0 + 2 h,... , x 0 +nh,... the scaled variable X takes on the values
X= 0, 1 , 2 ,... , n,....