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^\beta{0+} - {}{t_2}D^\gamma{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 ${}{t_1}D^\beta$ and ${}{t_1}D^\gamma$ are Caputo fractional derivatives of order $\beta, \gamma \in (0, 1)$ 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.