Discrete Fractional Calculus (2015)
3. Nabla Fractional Calculus
Christopher Goodrich1 and Allan C. Peterson2
(1)
Department of Mathematics, Creighton Preparatory School, Omaha, NE, USA
(2)
Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, USA
3.1. Introduction
As mentioned in the previous chapter and as demonstrated on numerous occasions, the disadvantage of the discrete delta fractional calculus is the shifting of domains when one goes from the domain of the function to the domain of its delta fractional difference. This problem is not as great with the fractional nabla difference as noted by Atici and Eloe. In this chapter we study the discrete fractional nabla calculus. We then define the corresponding nabla Laplace transform motivated by a particularly general definition of the delta Laplace transform that was first defined in a very general way by Bohner and Peterson [62]. Several properties of this nabla Laplace transform are then derived. Fractional nabla Taylor monomials are defined and formulas for their nabla Laplace transforms are presented. Then the discrete nabla version of the Mittag–Leffler function and its nabla Laplace transform is obtained. Finally, a variation of constants formula for an initial value problem for a ν-th, 0 < ν < 1, order nabla fractional difference equation is given along with some applications. Much of the work in this chapter comes from the results in Hein et al. [119], Holm [123–125], Brackins [64], Ahrendt et al. [3, 4], and Baoguo et al. [49, 52].