On bivariate fractional calculus with general univariate analytic kernels

We introduce a general bivariate fractional calculus, defined using a kernel based on an arbitrary univariate analytic function with an appropriate bivariate substitution. Various properties of the introduced general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. A major result of this paper is a fractional Leibniz rule for the new operators, the derivation of which involves correcting a minor error in one of the classic textbooks on fractional calculus. We also solve some fractional differential equations using transform methods, revealing an interesting connection between bivariate type Mittag-Leffler functions. •Bivariate fractional integrals using general analytic kernel functions.•A fractional Leibniz rule is proved for the new operators.•Corresponding bivariate fractional derivative operators are constructed.•Double Laplace transforms used to solve partial integro-differential equations.