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 ${}_{t_1}{D}^{\beta }$ 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 $k\ge 2$ number of fractional derivatives on the left hand side.

16 pages, 1 figure; updated references, minor typos and new figure