Abstract

The fractional Laplacian is a widely used tool in multi-dimensional fractional PDEs, useful because of its natural relationship with the multi-dimensional Fourier transform via fractional power functions. A well-known general class of fractional operators is given by fractional calculus with respect to functions; this has usually been studied in 1 dimension, but here we study how to extend it to an $n$-dimensional setting. We also formulate Fourier transforms with respect to functions, both in 1 dimension and in $n$ dimensions. Armed with these building blocks, it is possible to construct fractional Laplacians with respect to functions, both in 1 dimension and in $n$ dimensions. These operators can then be used for posing and solving some generalised families of fractional PDEs. Joint work with Joel E. Restrepo (Nazarbayev University) and Jean-Daniel Djida (AIMS Cameroon).