Analysis of a class of completely non-local elliptic diffusion operators

This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, \({D^\alpha _{a+}}{D^\beta _{b-}}\) $D_{a+}^{\alpha }{D}_{b-}^{\beta }$ , \(1<\alpha +\beta <2\) $1<\alpha +\beta <2$ . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of \({D^\alpha _{a+}}{D^\beta _{b-}}u(x)\) $D_{a+}^{\alpha }{D}_{b-}^{\beta }u(x)$ at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As \(\alpha \rightarrow 1^-\) $\alpha \to 1^{-}$ or \(\alpha ,\beta \rightarrow 1^-\) $\alpha ,\beta \to 1^{-}$ , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.