Physics of fractal operators

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This text explores how both deterministic and random fractal phenomena evolve over time through the lens of fractional calculus. It aims to pinpoint the characteristics of complex physical phenomena that necessitate fractional derivatives or integrals to accurately describe temporal changes. The focus is on physical phenomena whose evolution is best captured by fractional calculus, particularly systems exhibiting long-range spatial interactions or long-time memory. Traditional analytic function theory often falls short in modeling such complexities; thus, the text illustrates how less familiar functions, like fractals, can effectively serve as models. Notably, fractal functions, such as the Weierstrass function—which lacks a traditional derivative—can be represented through fractional derivatives and framed as solutions to fractional differential equations. The text also discusses how conventional differential equation-solving techniques, including Fourier and Laplace transforms and Green's functions, can be adapted to accommodate fractional derivatives. Furthermore, it outlines a comprehensive strategy for understanding wave propagation in random media, the nonlinear responses of complex materials, and the transport fluctuations in heterogeneous materials, while explaining the limitations of historical techniques as phenomena grow increasingly intricate.