What is Fractional Calculus?

Derivatives of non-integer order, also called fractional derivatives, have their origins slightly after the traditional calculus conception about 1695. However, the former had a late development due to its abstraction level and numerical requirements. These two issues have been fulfilled in recent decades, particularly researchers still pursuing to answer L’Hôpital’s question to Leibniz: “What does it mean if n = 1/2?”; since n is the order in Leibniz’s derivative notation.

Nowadays, it is common to refer to fractional calculus as the branch of mathematical analysis for studying properties and applications of the differential and integral operators of arbitrary order. It is easy to recognize two main alleys: The first is the development within pure mathematics, seldomly dealing with applications. In the second, the paradigm radically changes towards applications, for example, anomalous diffusion, power laws, allometric scaling laws, history-dependent phenomena, and rheology. It is worth mentioning that neither one nor another alley, the nonlocality is considered the most crucial characteristic.

Society is more connected every day, so non-local interactions are increasingly more predominant, especially in our world itself in these post-pandemic times. Such a fact will undoubtedly increase the number and depth of studies dedicated to the non-local models, which will develop analysis and experimentation in mathematical models through fractional calculus.

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