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.

这项工作探讨了将椭圆偏微分方程的一些经典结果模拟为一类分数常微分方程的可能性，涉及左、右黎曼-刘维尔 (RL) 分数导数的组合，\({D^\alpha _ {a+}}{D^\beta _{b-}}\) , \(1<\alpha +\beta <2\)。与单侧非局部 RL 导数相比，这些复合算子是完全非局部的，这意味着\({D^\alpha _{a+}}{D^\beta _{b-}}u 的评估(x)\)在点x处不仅必须检索x左侧一直到左边界的信息，而且还必须同时检索到右边界的信息。因此，只能使用有限的工具来应对这种情况，这是工作中最具挑战性的部分。为了克服这个问题，我们从非传统的角度进行分析，并最终建立椭圆型结果，包括霍普夫引理和极大值原理。作为\(\alpha \rightarrow 1^-\)或\(\alpha ,\beta \rightarrow 1^-\)，这些算子分别简化为单侧分数扩散算子和经典扩散算子。由于这些原因，我们仍然将它们称为“椭圆扩散算子”，但没有任何物理解释。