An analogue of Leibniz’s rule for Hadamard derivatives and their application

Abstract

This paper explores new analogues of the Leibniz rule for Hadamard and Caputo–Hadamard fractional derivatives. Unlike classical derivatives, fractional ones have a strong nonlocal character, meaning that the value of the derivative at a given point depends on the entire history of the function. Because of this nonlocality, the standard product rule cannot be directly applied. The study develops refined formulas for differentiating the product of two functions, which include additional integral terms representing memory effects inherent to fractional calculus. The paper also establishes a series of inequalities that make it possible to estimate the fractional derivatives of nonlinear expressions, such as powers of a function, through the derivative of the function itself. In particular, it is shown that a specific inequality holds for positive functions that relates the fractional derivative of the function power to the function product and its fractional derivative. These theoretical results are of great importance for the study of linear and nonlinear fractional diffusion equations. They provide useful tools for proving the existence, uniqueness, and stability of their solutions and for deriving a priori estimates that describe the qualitative behavior of such systems.