Vol 6, No 1

[Theme Issue] Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments, Part 1

Cover Page


M-Wright/Mainardi function. A wide variety of processes in engineering and applied science exhibit behaviour that cannot be modelled by classical methods, motivating and inspiring research on extended mathematical tools. Recently, the study of fractional calculus (integrals and derivatives of non-integer order) has led to new fundamental results. This approach has been successful in describing anomalous behaviour including fractional diffusion-wave equations. From a mathematical point of view, diffusion and wave equations are known to be governed by partial differential equations of order 1 and 2 in time. The introduction of time-fractional derivatives of real order beta ranging from 0 to 2 leads to fractional partial differential equations that well describe the desired anomalous behaviours. The cover image presents plots of the so-called M-Wright/Mainardi function M_nu(|x|) for different values of parameter nu that is associated to the order of derivation in time by the formula beta=2 nu. The M-Wright/Mainardi function embodies the Green function of the time-fractional diffusion-wave equation whose special cases are the Gaussian function, for pure diffusive processes nu=0.5, and the Dirac delta function, for pure wave motion nu=1. More details can be found in G. Pagnini, E. Scalas, Commun. Appl. Ind. Math., Vol 6 (1), e-496 (2015), doi:10.1685/journal.caim.496

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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909