Exact and Approximate Solutions of Fractional Partial Differential Equations for Water Movement in Soils

This paper presents solutions of the fractional partial differential equation (fPDE) for analysing water movement in soils. The fPDE explains processes equivalent to the concept of symmetrical fractional derivatives (SFDs) which have two components: the forward fractional derivative (FFD) and backward fractional derivative (BFD) of water movement in soils with the BFD representing the micro-scale backwater effect in porous media. The distributed-order time-space fPDE represents water movement in both swelling and non-swelling soils with mobile and immobile zones with the backwater effect operating at two time scales in large and small pores. The concept of flux-concentration relation is now updated to account for the relative fractional flux of water movement in soils.