A comparative study on using metaheuristic algorithms for simultaneously estimating parameters of space fractional advection-dispersion equation

Abstract

This work developed four inverse models based on Particle Swarm Optimization (PSO), Chicken Swarm Optimization (CSO), Bees Algorithm (BA), and Teaching Learning Based Optimization (TLBO), to identify parameters of space fractional advection–dispersion equation (s-FADE). The s-FADE has four parameters, including average pore-water velocity (v), fractional dispersion coefficient (Df), fractional derivative order (α), and skewness (β). A sensitivity analysis indicated that the v is the most effective parameter on the s-FADE results, followed by α, Df, and β, respectively. The experimental data required were measured at different transport distances of homogeneous and heterogeneous soil columns. Five criteria, namely, convergence trend, objective function value, runtime, repeatability of results, and modeling complexity were used to evaluate algorithm performances and to rank them using a technique for order of preference by similarity to ideal solution (TOPSIS). Based on the obtained results, all four algorithms acquired the global optimal values for the v and α parameters using a maximum iteration of 1000 as a stopping criterion and an initial population of 10, while they obtained the relatively different values for the Df, and β parameters. The PSO and TLBO algorithms successfully found the global minimum values of the objective functions for both the homogeneous and heterogeneous soils. Among the four algorithms, the TLBO algorithm was the best one in terms of convergence trend, repeatability of results, and modeling complexity, and it was the worst algorithm in term of runtime. Among the PSO, CSO, and BA algorithms, the BA algorithm was superior over the PSO and CSO algorithms in terms of runtime and repeatability of results, while the PSO algorithm was superior over the BA and CSO algorithms in term of converge speed. Overall, according to the results of the TOPSIS method, the TLBO algorithm was the best alternative to estimate the s-FADE parameters, followed by BA, PSO, and CSO algorithms. Also, the comparison of the s-FADE parameters estimated by the TLBO algorithm, as the best one among the four algorithms, with those estimated by FracFit toolbox revealed that both the techniques obtained the relatively identical and admissible values for the v and α parameters, while the TLBO algorithm acquired the more precise values for the Df and β parameters. A detailed analyses demonstrated that the TLBO algorithm was markedly superior to the FracFit toolbox in terms of the aforementioned criteria. In a nutshell, the TLBO algorithm can be used as a highly efficient optimization method to estimate the s-FADE parameters in both the homogeneous and the heterogeneous soils.