Mixed linear fractional boundary value problems

Abstract

In this article we obtain two-sided estimates for the Greens function of fractional boundary value problems on $\mathbb R_+ \times \mathbb R_+ \times \mathbb R^d$ of the form \[(-{}_{t_1}D^β_{0+*} - {}_{t_2}D^γ_{0+*})u(t_1, t_2, x) = L_{x}u(t_1, t_2, x),\] with some prescribed boundary functions on the boundaries $\{0\} \times \mathbb R_+ \times \mathbb R^d$ and $\mathbb R_+ \times\{0\}\times \mathbb R^d$. The operators ${}_{t_1}D^β$ and ${}_{t_1}D^γ$ are Caputo fractional derivatives of order $β, γ\in (0, 1)$ and $L_{x}$ is the generator of a diffusion semigroup: $L_x= \nabla \cdot(a(x) \nabla)$ for some nice function $a(x)$. The Greens function of such boundary value problems are decomposed into its components along each boundary, giving rise to a natural extension to the case involving $k \geq 2$ number of fractional derivatives on the left hand side.