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How to impose physically coherent initial conditions to a fractional system? (English) Zbl 1221.34019

Summary: It is shown that neither Riemann–Liouville nor Caputo definitions for fractional differentiation can be used to take into account initial conditions in a convenient way from a physical point of view. This demonstration is done on a counter-example. Then the paper proposes a representation for fractional order systems that lead to a physically coherent initialization for the considered systems. This representation involves a classical linear integer system and a system described by a parabolic equation. It is thus also shown that fractional order systems are halfway between these two classes of systems, and are particularly suited for diffusion phenomena modelling.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
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