Modeling with Difference Equations

Abstract

In this chapter, we return to the questions of how difference equations are formulated and how difference equations arise. We will proceed by example and attempt to give only a broad sample of the many uses of difference equations. Whenever appropriate, the methods of the previous chapters will be applied to solve the various equations. Of course, we have added here a new feature for this book, namely, the use of the discrete Fourier transforms for solving difference equations associated with boundary conditions, especially their direct use in algebraizing the difference equation and at the same time accommodating the boundary conditions. The z-transform has long been used in difference equations associated with initial conditions. Thus, the direct use of the discrete Fourier transforms in this book plus the use of the z-transform constitute examples of what we term, the “operational sum calculus” method. We may add that more “modified” discrete Fourier and the more general “Sturm—Liouville” — type discrete transforms may be constructed in parallel to what was done for their parallels of the general finite transforms that were discussed briefly in Section 2.3.