The Green's function and a maximum principle for a Caputo two-point boundary value problem with a convection term

A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α∈(1,2) and where a convection term is also present is considered. Its boundary conditions are of Robin type and include Dirichlet boundary conditions as a special case. An explicit formula for the associated Green's function is obtained in terms of two-parameter Mittag-Leffler functions. Some new properties of these Mittag-Leffler functions are derived; from these, one can deduce necessary and sufficient conditions on the boundary conditions that ensure non-negativity of the Green's function and hence a maximum principle for the boundary value problem. In particular, unlike the classical elliptic case α=2, Dirichlet boundary conditions will not in general yield a maximum principle.