Mikusiński’s operational calculus for multi-dimensional fractional operators with applications to fractional PDEs

Abstract

We construct, for the first time, a Mikusiński-type operational calculus structure for partial differential operators of non-integer order. Our operators are of Riemann–Liouville type, and in arbitrary dimensions, although we often focus on the two-dimensional case as a model problem. We establish suitable function spaces, algebraic properties, and interpretations of multi-dimensional fractional integral and derivative operators. As an example application, we consider a fractional differential equation in two dimensions posed on the first quadrant, and find its explicit solution using Wright functions.