Bivariate substitutions from analytic kernels to fractional differintegral operators

Abstract

We study a general class of bivariate fractional integral operators, derived from bivariate analytic kernel functions by a double substitution of variables and a double convolution. We also study the associated bivariate fractional derivative operators, derived by combining partial derivatives with the aforementioned fractional integrals. These operators give rise to a theory of two-dimensional fractional calculus which is general enough to include many existing models involving different kernel functions with applications.