Modeling and simulation of the fractional space-time diffusion equation

In this paper, the space-time fractional diffusion equation related to the electromagnetic transient phenomena in transmission lines is studied, three cases are presented; the diffusion equation with fractional spatial derivative, with fractional temporal derivative and the case with fractional space-time derivatives. For the study cases, the order of the spatial and temporal fractional derivatives are 0 < β, γ ≤ 2, respectively. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional diffusion equation. The general solutions of the proposed equations are expressed in terms of the multivariate Mittag-Leffler functions; these functions depend only on the parameters β and γ and preserve the appropriated physical units for any value of the fractional derivative exponent. Furthermore, an analysis of the fractional time constant was made in order to indicate the change of the medium properties and the presence of dissipation mechanisms. The proposed mathematical representation can be useful to understand electrochemical phenomena, propagation of energy in dissipative systems, irreversible thermodynamics, quantum optics or turbulent diffusion, thermal stresses, models of porous electrodes, the description of gel solvents and anomalous complex processes.