Fractional Derivatives: an extension of classical analysis to non-integer orders
Félix del Teso, David Gómez-Castro
TL;DR
The paper introduces fractional calculus by detailing three core operator families—Riemann–Liouville, Caputo, and Grünwald–Letnikov—and shows how they extend classical derivatives and integrals with memory effects via kernels and Mittag–Leffler functions. It develops the fundamental theorems linking fractional derivatives and integrals, derives solution forms for linear and nonlinear fractional ODEs (including Caputo and RL cases) through Mittag–Leffler functions, and presents variation-of-constants formulas alongside Laplace-transform techniques. The text then extends these ideas to systems of equations and outlines numerical methods suitable for both local and fractional problems, highlighting the nonlocal time dependence that arises in fractional settings. Two illustrative applications—the tautochrone problem reformulated with fractional derivatives and the Rough Heston model in finance—demonstrate the practical impact of fractional calculus in physics and quantitative finance. Overall, the work provides a structured, accessible account of fractional operators, their interrelations, solution frameworks, numerical approaches, and real-world applications, emphasizing memory effects and generalized dynamics.
Abstract
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant properties and applications of this mathematical tool. It was originally published in Spanish in the Gaceta de la Real Sociedad Española and automatically translated using Github copilot.