Discrete Transform Methods of Solutions of the Fractional Difference Equations

The fractional difference equations are encountered in many areas of applications, notably in control theory and control engineering, combinatorics, algebra, and mathematical analysis. In order to solve fractional differential equations conveniently, in this paper, the operational properties of fractional difference are studied. We begin with an introduction to the definitions of Delta left fractional difference of order α > 0, Nabla left fractional difference of order α > 0, Delta left fractional sum, and Nabla left fractional sum. Then we introduce the definitions and properties of Discrete R-Transform and Discrete N-transform. Finally, using the properties of fractional differential operation and discrete variation methods we obtain solutions of a class of fractional difference equations.