Difference equations

Key Words: Difference Equation, Solution Function, Sequence, Generating Function This research is based on the book named ?Introduction to Difference Equations? by Samuel Goldberg. In this study, properties of the linear difference equations and their solutions are investigated. Various applications in different diciplines are emphasized. In the first section it is illustrated that, problems that can take a discrete set of possible values often lead to mathematical models involving difference equations. In the second, difference function, some properties of the D operator and similarities between difference and differential are presented. The third section includes solutions of the first-order difference equations, their existence and uniqueness, presentation of these solutions by series and also their limiting behaviour. Additionally, some approximations have been made to the differential equations by difference equations. The topic of the fourth section is homogeneous and non-homogeneous difference equations with constant coefficient. In this part, behaviours of the solutions in the limiting case are detailed. In addition to the general case of order n, the relationships with the linear differential equations with constant coefficients are displayed. In the fifth section, equilibrium and stability of a difference equation is defined and the way they depend on the initial conditions is sampled. Relationships between the first-order equations and the cobweb cycles are introduced, a second-order boundaryvalue problem is defined and its non-trivial solutions are searched. Finally, I this section the generating function is defined and is application on different problems is pointed out.