Variational formulations of differential equations and asymmetric fractional embedding

Variational formulations for classical dissipative equations, namely friction and diffusion equations, are given by means of fractional derivatives. In this way, the solutions of those equations are exactly the extremal of some fractional Lagrangian actions. The formalism used is a generalization of the fractional embedding developed by Cresson [Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007) 033504], where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives. Moreover, this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle.