Application of Fractional Order Calculus in Physics

The concept of fractional calculus (fractional derivative and fractional integral) is not new. In recent years, fractional calculus has found use in studies of viscoelastic materials, as well as in many fields of science and engineering including fluid flow, rheology, diffusive transport, electrical networks, electromagnetic theory and probability. The definitions of Fractional Derivatives and Integrals, that I chose to consider: Grunwald-Letnikov Fractional Derivatives; Riemann-Liouville Fractional Integrals; Riemann-Liouville Fractional Derivatives; Caputo Fractional Derivatives. We take the phenomenological diffusion equation and study the different cases leading to anomalous diffusion. Modelling the memory effects of anomalous diffusion, we get the fractional time form of Cattaneo equation, with 0 < ɑ < 1 and D is the diffusion constant with τ<< 1. If ɑ= 1, the normal Cattaneo diffusion equation is recovered. We try to model the fractional diffusion equations for anomalous diffusion cases and simulate the diffusion phenomena.