Solutions of Nabla Fractional Difference Equations Using N-Transforms

Abstract

In the present paper, we present some important properties of N-transform, which is the Laplace transform for the nabla derivative on the time scale of integers (Bohner and Peterson in Dynamic equations on time scales, Birkhauser, Boston, 2001; Advances in dynamic equations on time scales, Birkhauser, Boston, 2002). We obtain the N-transform of nabla fractional sums and differences and then apply this transform to solve some nabla fractional difference equations with initial value problems. Finally, using N-transforms, we prove that discrete Mittag-Leffler function is the eigen function of Caputo type nabla fractional difference operator \(\nabla ^{\alpha }\).