Modelling of the Convection-Diffusion Equation through Fractional Restricted Calculus of Variations.

Abstract

We deliver a restricted variational principle that provides, after the application of the Hamilton’s principle, a particular set of α-fractional PDEs as sufficient condition for the extremals when it is applied over a given action. This action is defined as the integral of a particular Lagrangian function containing the fractional derivatives (both for space and time variables) of the involved scalar fields as part of the state space. We see that the convection-difussion equation is a particular instance of these α-fractional PDEs, say when α = 1/2. Furthermore, we briefly explore some paths towards discretisation of the restricted variational principle and the fractional PDEs. It is to be expected that the obtained discrete version of the equations, coming from a variational principle, would inherit some of the geometric/preservation properties of their continuous counterparts. This is left as subject of future study.