Green's function for higher-order boundary value problems involving a nabla Caputo fractional operator

TL;DR

This paper develops a Green's function approach for higher-order boundary value problems involving a nabla Caputo fractional operator, extending classical methods to fractional difference equations.

Contribution

It introduces a Green's function framework for fractional difference boundary value problems with a nabla Caputo operator, including solution formulas and specific boundary cases.

Findings

Established properties of solutions to fractional difference equations.

Derived a variation of constants formula using a Cauchy function.

Defined Green's function and proved Green's Theorem for these problems.

Abstract

We consider the discrete, fractional operator $\left(L_a^\nu x\right) (t) := \nabla [p(t) \nabla_{a^*}^\nu x(t)] + q(t) x(t-1)$ involving the nabla Caputo fractional difference, which can be thought of as an analogue to the self-adjoint differential operator. We show that solutions to difference equations involving this operator have expected properties, such as the form of solutions to homogeneous and nonhomogeneous equations. We also give a variation of constants formula via a Cauchy function in order to solve initial value problems involving $L_a^\nu$. We also consider boundary value problems of any fractional order involving $L_a^\nu$. We solve these BVPs by giving a definition of a Green's function along with a corresponding Green's Theorem. Finally, we consider a (2,1) conjugate BVP as a special case of the more general Green's function definition.