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

2018/10/10 by Ahrendt, Kevin, Kissler, Cameron
#39A10 #39A70 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.04628

Abstract

We consider the discrete, fractional operator (L_a^νx) (t) := ∇ [p(t) ∇_a^*^νx(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^ν. We also consider boundary value problems of any fractional order involving L_a^ν. 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.

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