Abstract
AbstractThis 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
+
α
D
b
-
β
, $$1<\alpha +\beta <2$$
1
<
α
+
β
<
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
+
α
D
b
-
β
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 $$\alpha \rightarrow 1^-$$
α
→
1
-
or $$\alpha ,\beta \rightarrow 1^-$$
α
,
β
→
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.
Publisher
Springer Science and Business Media LLC