Initial value problems in discrete fractional calculus

Atici, Ferhan; Eloe, Paul

doi:10.1090/S0002-9939-08-09626-3

Initial value problems in discrete fractional calculus
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by Ferhan M. Atici and Paul W. Eloe PDF

Proc. Amer. Math. Soc. 137 (2009), 981-989 Request permission

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 $\nu$-th ($0 < \nu \leq 1$) 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.

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