Initial value problems in discrete fractional calculus

Abstract

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a -th (<!-- MATH $0 < \nu \leq 1$ --> <img width="90" height="37" align="MIDDLE" border="0" src="/proc/2009-137-03/S0002-9939-08-09626-3/gif-abstract0/img2.gif" alt="$ 0 < \nu \leq 1$">) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.