Approximate analytical solutions for multispecies convection-dispersion transport equation with variable parameters

Abstract

Multispecies pollutant migration often occurs in polluted groundwater systems. Most of the multispecies problems that have been dealt in the literature assume constant transport parameters, primarily because analytical solutions for varying parameters become a challenge. The present study analytically solves a two-species convection-dispersion transport equation, considering spatially varying dispersion coefficient and seepage velocity, which corresponds to the steady migration in a steady flow domain. Indeed, the methodology can be adopted for other cases, such as the transient migration in a steady flow domain and transient migration in an unsteady flow domain, without any difficulty. Three kinds of homotopy-based methods, namely the homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimal homotopy asymptotic method (OHAM), are used to derive approximate analytical solutions in the form of truncated series. In homotopy analysis method, the convergence-control parameter ℏ plays a key role in the convergence of the series solution. It is observed that for a specific case of this parameter, namely ℏ=−1, the HAM-based solution recovers the HPM-based solution. For the verification of homotopy-based solutions, we utilize the MATLAB routine pdepe, which efficiently solves a class of parabolic PDEs (single/system). An excellent agreement is found between the derived analytical solutions and the numerical solutions for all three methods. Further, a quantitative assessment is carried out for the derived solutions. Also, convergence theorems are proposed for the series solutions obtained using HAM and OHAM.