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This Special Issue is devoted to some serious problems that the Fractional Calculus (FC) is currently confronted with and aims at providing some answers to the questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?’’, etc. In particular, the “new fractional derivatives and integrals” and the models with these fractional order operators are critically addressed. The Special Issue contains both the surveys and the research contributions. A part of the articles deals with foundations of FC that are considered from the viewpoints of the pure and applied mathematics, and the system theory. Another part of the Special issue addresses the applications of the FC operators and the fractional differential equations. Several articles devoted to the numerical treatment of the FC operators and the fractional differential equations complete the Special Issue.

Research & information: general --- Mathematics & science --- fractional derivatives --- fractional integrals --- fractional calculus --- fractional anti-derivatives --- fractional operators --- integral transforms --- convergent series --- fractional integral --- fractional derivative --- numerical approximation --- translation operator --- distributed lag --- time delay --- scaling --- dilation --- memory --- depreciation --- probability distribution --- fractional models --- fractional differentiation --- distributed time delay systems --- Volterra equation --- adsorption --- fractional differential equations --- numerical methods --- smoothness assumptions --- persistent memory --- initial values --- existence --- uniqueness --- Crank–Nicolson scheme --- weighted Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- stability analysis --- convergence order --- Caputo–Fabrizio operator --- Atangana–Baleanu operator --- fractional falculus --- general fractional derivative --- general fractional integral --- Sonine condition --- fractional relaxation equation --- fractional diffusion equation --- Cauchy problem --- initial-boundary-value problem --- inverse problem --- fractional calculus operators --- special functions --- generalized hypergeometric functions --- integral transforms of special functions