An analogue of Leibniz’s rule for Hadamard derivatives and their application
Smadiyeva A.G.
30 December 2025E.A. Buketov Karaganda University Publish house
Bulletin of the Karaganda University. Mathematics Series
2025#2025Issue 4180 - 195 pp.
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.
a priori estimates , Gronwall inequality , Hadamard time fractional derivative , Hadamard-type time fractional derivative , Leibniz rule , linear and nonlinear diffusion equation , Mittag-Leffler function , porous medium equation
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Al–Farabi Kazakh National University, Al–Farabi 71,, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Shevchenco 28, Almaty, Kazakhstan
Al–Farabi Kazakh National University
Institute of Mathematics and Mathematical Modeling
10 лет помогаем публиковать статьи Международный издатель
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