09-20-2026, 05:42 PM
Fractional calculus is a generalization of ordinary calculus in which differentiation and integration can be performed to non-integer orders, such as $1/2$, $3/2$, or even complex orders. Instead of considering only $D^n f$ for integer $n$, it asks what an operator such as $D^{1/2}$ might mean—essentially, whether one can define a half derivative.
The idea dates back to correspondence between Leibniz and l’Hôpital in 1695 and was developed mathematically by researchers including Abel and Liouville. Fractional integration can be constructed naturally by extending the factorial in repeated integration to the Gamma function, leading to the Riemann–Liouville fractional integral and the important semigroup property $J^\alpha J^\beta=J^{\alpha+\beta}$.
Fractional derivatives are considerably more complicated because there is no single universally accepted definition. Major formulations include the Riemann–Liouville, Caputo, Caputo–Fabrizio, Atangana–Baleanu, and Riesz derivatives, which can produce different results even for smooth functions. Fractional differential equations extend ordinary differential equations and are useful for describing systems with memory, nonlocal effects, anomalous diffusion, damping, control systems, acoustic waves, groundwater flow, and certain quantum-mechanical models. The central idea is that fractional operators provide a mathematical way to describe processes that lie between purely local differentiation and integration and more complicated long-memory behavior.
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The idea dates back to correspondence between Leibniz and l’Hôpital in 1695 and was developed mathematically by researchers including Abel and Liouville. Fractional integration can be constructed naturally by extending the factorial in repeated integration to the Gamma function, leading to the Riemann–Liouville fractional integral and the important semigroup property $J^\alpha J^\beta=J^{\alpha+\beta}$.
Fractional derivatives are considerably more complicated because there is no single universally accepted definition. Major formulations include the Riemann–Liouville, Caputo, Caputo–Fabrizio, Atangana–Baleanu, and Riesz derivatives, which can produce different results even for smooth functions. Fractional differential equations extend ordinary differential equations and are useful for describing systems with memory, nonlocal effects, anomalous diffusion, damping, control systems, acoustic waves, groundwater flow, and certain quantum-mechanical models. The central idea is that fractional operators provide a mathematical way to describe processes that lie between purely local differentiation and integration and more complicated long-memory behavior.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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