Analytical Solutions of One-Dimensional Space-Fractional Advection–Diffusion Equation for Sediment Suspension Using Homotopy Analysis Method

Abstract

In this study, the full analytical solution of one-dimensional space-fractional advection–diffusion equation (fADE) is derived using the homotopy analysis method (HAM). Apart from the several applications of the fADE, it has been applied in sediment-laden turbulent flows to find the distribution of suspended sediment particles when the effect of nonlocal transport is present. In such flows, due to the presence of turbulent-bursting particles ejected from the bed level up to the free surface performing nonlocal vertical jumps, which cannot be captured through traditional integer-order concentration gradient, the fractional derivative of concentration, therefore, comes into promise. The fractional derivative considered here contains a convolution of suspension concentration and concentration gradient which captures the effect over all points throughout the domain. The HAM is a powerful method to solve the nonlinear fractional differential equations and has been applied here to obtain the solutions. The advantage of this proposed HAM solution over previous analytical solutions of fADE, in terms of the Mittag-Leffler function, is that it allows to choose the region of convergence of the series solution with greater freedom, and thus broadens the applicability of the model under many difficult situations. The HAM solutions are obtained for two different choices of sediment diffusivity and they are also compared with analytical solution and numerical solutions. The solutions are also validated with the experimental data. The comparison results are satisfactory and it shows the advantage of using the homotopy analysis method. Finally, the variations of the model with the parameters are also presented and the reasons are justified physically.