From a Fractional Mechanical Model to a Fractional Generalization of the Dirac Equation

摘要

The Fractional Calculus represents a natural instrument to model non-local phenomena either in space or time that involve different scales. In this paper, we present a generalization of the linear one-dimensional diffusion and wave equations obtained by combining the fractional derivatives and the internal degrees of freedom associated to the system. Actually, taking into account that the free Dirac equation is, in some sense, the square root of the Klein-Gordon equation, in a similar way we can operate a kind of square root of the time fractional Diffusion equation in one space dimension through the system of fractional evolution-diffusion equations Dirac like. Solutions of the above system could model the diffusion of particles whose behavior depends not only on the space and time coordinates, but also on their internal structures. We study some analytical and invariance properties of the system highlighting its interpolation between the hyperbolic and parabolic behaviors.