This work presents the first attempt to develop unconditionally stable, implicit finite difference solutions of one-sided spatial fractional advection-dispersion equation (s-FADE) by imposing the nonzero Dirichlet boundary condition (ND BC) or the nonzero fractional Robin boundary condition (NFR BC) at inlet boundary and the zero fractional Neumann boundary condition (ZFN BC) at outlet boundary. The results of the numerical studies performed using artificial solute transport parameters demonstrated that the numerical solution with the NFR BC as the inlet boundary produced much more realistic concentration values. The numerical solution with the NFR BC at the inlet boundary was capable of correctly describing the Fickian and non-Fickian behaviors of the solute transport at different α values, and it had the relatively same accuracy at different numbers of the spatial nodes. Also, the practical application of the numerical solution with the NFR BC as the inlet boundary was investigated by conducting tracer experiments in homogeneous and heterogeneous soil columns. According to the obtained results, this numerical solution described well solute transport in the homogenous and heterogeneous soils. The α values of the homogeneous and heterogeneous soils were obtained in the ranges of 1.849 to 1.999 and 1.248 to 1.570, respectively, which were in excellent agreement with the physical properties of the soils. In a nutshell, the numerical solution of the s-FADE with the NFR BC as the inlet boundary can be successfully applied to describe the solute transport in the homogeneous and heterogeneous soils with bounded spatial domains.
