Semi-analytical solutions for two-dimensional convection-diffusion-reactive equations based on homotopy analysis method.

Convention applied to describe contaminant transport in landfills and groundwater systems is typically characterized by simplified geometries and boundary conditions. As a result, they neglect the more general boundary conditions encountered in the real world, including convection and diffusion of contaminants (e.g., landfill leachate) associated with fluid transportation in the lateral direction. Here, we present semi-analytical solutions that can be used to describe and estimate the contaminants' fate in two-dimensional space. This is achieved by applying the homotopy analysis method (HAM) to create a different order deformation equation series, the sum of which is the solution of the two-dimensional target problem. To ensure the accuracy of the semi-analytical solution, elements of the equation series have been defined and adapted to satisfy the partial differential equation of the discussed problem. Similarly, the convergence of the HAM solution has been achieved by adopting proper convergent control parameters, ensuring the convergence of each element of the deformation equation series. This guarantees that the sum of the equation series is convergent. HAM has been applied to three cases with more general and smooth initial conditions. Good agreement between HAM solutions and numerical solutions from the literature demonstrates the capacity of HAM.