TABLE OF CONTENTS Difference equations arise in both the modeling of discrete dynamical systems and in various purely mathematical contexts, including the numerical integration of differential equations. This chapter begins with the genesis of several difference equations and is followed by a section giving the conditions under which existence and uniqueness of solutions occur. Next the fundamental "difference" and "shift" operators are defined and their important properties derived. Sections 5.5 and 5.6 study first-order and general linear difference equations. A number of theorems are given and we show how they can be applied to determine solutions for these types of difference equations. In section 5.8, the details as to constructing general solutions to linear equations with constant coefficients are presented. Both homogeneous and inhomogeneous equations are considered, with examples to illustrate the application of the stated methods.
Nonlinear difference equations are treated in section 5.9, with worked examples in section 5.10. We only consider the cases of several special classes of nonlinear equations. They have the property of being transformed, by a change of dependent variable, into linear equations that can be solved. Finally, we give, in section 5.11, two special applications of the use of difference equations. The first is related to the Chebyshev polynomials, the second is concerned with the construction and analysis of discrete models for the logistic differential equation.
It will be noticed that there is a great similarity between the theorems of difference and differential equations [1]; [2]; [3] and...
