Local fractional Moisil–Teodorescu operator in quaternionic setting involving Cantor-type coordinate systems

The Moisil-Teodorescu operator is considered to be a good analogue of the usual Cauchy–Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in (Formula presented.). In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu operator in Cantor-type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil–Teodorescu operator.