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

Abstract

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, \({D^\alpha _{a+}}{D^\beta _{b-}}\) , \(1<\alpha +\beta <2\) . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of \({D^\alpha _{a+}}{D^\beta _{b-}}u(x)\) 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 \(\alpha \rightarrow 1^-\) or \(\alpha ,\beta \rightarrow 1^-\) , 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.