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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. In this work, 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's functions. We investigate properties of the Green's functions and explore some useful generalizations involving sequential nabla fractional differences.
Setniker, Ariel, "Sequential Differences in Nabla Fractional Calculus" (2019). ETD collection for University of Nebraska - Lincoln. AAI13861982.