Course Summary

Difference equations appear naturally in the description of observed phenomena and their evolution since the typical data from measurements is discrete. Moreover, they also appear in the discretization methods for differential, integral and integro-differential equations, which appear as continuous models for the same discrete measurements, using computers which manipulate the unknown function only at a discrete set. The theory and application of difference equations covers a diverse variety of engineering and scientific fields which are rapidly increasing including specific areas such as particle dynamics, numerical analysis, control theory, dynamical systems, special functions, combinatorics, integrable systems, with general applications to mechanics, computer science, biology, physics, etc.

Difference equations are to some extent analogous to differential equations. Whereas a differential equation involves an unknown function defined on a continuous domain, a difference equation involves an unknown sequence which is defined on a discrete set. The differential equations are slightly harder to comprehend as the notion of a derivative is based on a limiting process. By contrast, difference equations, in some sense, are relatively easy to grasp. More so with the aid of a computer algebra system, programs such as Mathematica, Maple, assist the students to experiment with difference equations, perform analysis of such equations, and discover their structure and properties.