Sequential Differences in Nabla Fractional Calculus

We study the composition of nabla fractional differences of unequal orders, known as sequential nabla fractional differences. The sequential differences we examine possess different bases — specifically, we establish the outer operator as having a base larger than the inner operator by at least an integer factor of 1. Further, we consider two cases of orders: first the case when the outer difference has a larger power, and second when the inner difference has a larger power. We develop rules for sequential nabla fractional differences and present connections between the sign of a sequential difference of a function and the monotonicity of that function. We establish uniqueness of solutions to various initial value problems and boundary value problems involving sequential nabla fractional differences and give an explicit expression for the Green\u27s functions. We investigate properties of the Green\u27s functions and explore some useful generalizations involving sequential nabla fractional differences. Adviser: Allan C. Peterso

A Coupled System of Fractional Difference Equations with Nonlocal Fractional Sum Boundary Conditions on the Discrete Half-Line

In this article, we propose a coupled system of fractional difference equations with nonlocal fractional sum boundary conditions on the discrete half-line and study its existence result by using Schauder’s fixed point theorem. An example is provided to illustrate the results