NUMERICAL STUDY OF FRACTIONAL EVOLUTION-DIFFUSION EQUATIONS DIRAC LIKE

Through the Fractional Calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon one, a possible interpolation between the Dirac, diffusion and wave equations in one space dimension can be derived, that we named fractional evolution-diffusion equation Dirac like. Such an equation contains a fractional derivative of order α varying in (0, 1] in time and a first order partial derivative in space. It can be seen as one of the two roots that we would obtain operating a kind of square root of the time fractional diffusion equation in one space dimension, with fractional derivative in time of order α [1, 2]. Solutions of this equation could model the diffusion of particles whose behavior depends not only on the space and time coordinates, but also on their internal structures. A numerical scheme based on convolution quadrature formula is given for solving this equation and the associated stability bounds are checked in some concrete case.