Fractional Calculus: A 300-Year Perspective on Operators, Applications, and Modern Generalizations

Resumo

This work provides a comprehensive historical and analytical review of fractional calculus, tracing its evolution from the seminal concepts of Leibniz (1695) to modern unified local and biparametric generalizations. The study highlights two main historical branches: the classical nonlocal operators (Riemann--Liouville, Caputo, and Riesz), which emphasize long-range memory effects crucial for modeling viscoelasticity and anomalous diffusion; and the later emergence of local fractional derivatives (Yang, Conformable), designed to capture local scaling properties in fractal and non-smooth media.

Key milestones include the introduction of the Caputo derivative in 1967, which facilitated the use of classical initial conditions in fractional differential equations, and the development of variable-order operators (2000s) to model systems with dynamic memory. The paper critically examines modern trends, focusing on the Unified Local Fractional Derivative (ULFD) and the recent Biparametric V-Derivative.

This derivative offers an enhanced flexible framework by independently tuning the local scaling exponent ($\alpha$) and the classical interpolation ($\beta$). Furthermore, the associated generalized operational calculus, which provides closed-form Laplace transform solutions, is discussed. We conclude that contemporary research is focused on developing hybrid formalisms that unify nonlocal memory, local scaling, and classical dynamics, offering superior flexibility for modeling complex phenomena like hybrid transport and fractal media