Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models

摘要

Fractional calculus,which has two prominent characteristics—singularity and nonlocality,comprises the integration and differentiation of any positive real (and even complex) order.It has a more than three-hundred-year history and can be traced back to a letter from Leibniz to L'H(o)pital,dated 30 September 1695,in which the meaning of the one-half order derivative was first discussed and some remarks about its feasibility were made.Abel was probably the first who rendered an application of fractional calculus.He used the derivatives of arbitrary order to solve the tautochrone (isochrone) problem in 1823.Fractional calculus underwent two periods:from its beginning to the 1970s,and after 1970s.During the first period,fractional calculus was studied mainly by mathematicians as an abstract field containing only pure mathematical manipulations of little applications,except for sporadic applications in rheology.During the second period,the paradigm shifted from pure mathematical research to applications in various realms,including anomalous diffusion,anomalous convection,power laws,allometric scaling laws,history dependence,longrange interactions,and so on.