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.

Automated summary

We present a general bivariate fractional calculus method using a kernel based on an arbitrary univariate analytic function. Various properties of the general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. Notably, we derive a fractional Leibniz rule for the new operators and correct a minor error in a classic textbook on fractional calculus. Additionally, we solve fractional differential equations using transform methods and uncover an interesting connection between bivariate Mittag-Leffler functions.