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Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding.

fractional diffusion --- continuous time random walks --- reaction–diffusion equations --- reaction kinetics --- multidimensional scaling --- fractals --- fractional calculus --- financial indices --- entropy --- Dow Jones --- complex systems --- Skellam process --- subordination --- Lévy measure --- Poisson process of order k --- running average --- complexity --- chaos --- logistic differential equation --- liouville-caputo fractional derivative --- local discontinuous Galerkin methods --- stability estimate --- Mittag-Leffler functions --- Wright functions --- fractional relaxation --- diffusion-wave equation --- Laplace and Fourier transform --- fractional Poisson process complex systems --- distributed-order operators --- viscoelasticity --- transport processes --- control theory --- fractional order PID control --- PMSM --- frequency-domain control design --- optimal tuning --- Gaussian watermarks --- statistical assessment --- false positive rate --- semi-fragile watermarking system --- fractional dynamics --- fractional-order thinking --- heavytailedness --- big data --- machine learning --- variability --- diversity --- telegrapher’s equations --- fractional telegrapher’s equation --- continuous time random walk --- transport problems --- fractional conservations laws --- variable fractional model --- turbulent flows --- fractional PINN --- physics-informed learning --- n/a --- reaction-diffusion equations --- Lévy measure --- telegrapher's equations --- fractional telegrapher's equation