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2026-03-09
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Abstract
Fractional calculus generalizes the concepts of differentiation and integration by extending their orders from integer to real or complex numbers, thereby permitting the exploration of fractional-order dynamics. Rooted in the pioneering thoughts of Leibniz and Euler in the late 17th and early 18th centuries, fractional calculus has undergone substantial theoretical and methodological developments, especially in recent decades. Its relevance arises from the unique ability to represent memory effects, non-local interactions, and hereditary properties that classical integer-order calculus fails to adequately capture. These attributes make fractional calculus exceptionally suited to modeling phenomena characterized by complexity and irregularities, including anomalous diffusion processes, viscoelastic materials, turbulence, and fractal media.
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Numerical Methods in Porous Media MATLAB® and Python Approaches