This study aims at investigating the numerical analysis of pollutant transport in homogenous porous media in the presence of plate stacks. The investigation is performed for solid objects of the same size placed in an inline arrangement in a homogenous porous media. The pollutant transport equation (i.e., steady-state and time-dependent advection-diispersion) is chosen in mathematical modeling. Furthermore, on basis of dispersion coefficient, three more cases arise which include uniformly constant, dependence on the magnitude of velocity, and dependence on both magnitude of velocity and its direction. Such models have a wide range of applications. Generally, the analytical solution of such problems doesn’t exist, so all the work is done numerically. The governing partial differential equation of pollutant concentration is approximated by using finite difference technique. Central and one-sided finite difference formulae are used to discretize the domain. MATLAB software is used to compute approximations to velocity potential and stream function. Then equipotential lines and streamlines are visualized in form of contours. Both, velocity potential and stream function satisfy Laplace’s equation and they are harmonic. Fluid flow lines and pollutant concentration are represented graphically for various parameters involved. It is observed that the size, shape, and position of pervious objects, entrance length of the domain affect fluid flow and pollutant transport. However, there is no significant effect of heated objects on pollutant transport. Moreover, advection and dispersion depend upon the permeability of porous media and the properties of the solid matrix.

In this study, an innovative analytical analysis of fractional-order partial differential equations is presented by Shehu transformation method. Fractional-order differential equations provide the useful dynamics of phys- ical systems and thus provide novel and efficient information about given physical systems. In this study, Shehu transform is used to create an approximate analytical solution through the time-fractional partial differential equations (system of equations) with the Adomian decomposition method and Variational iter- ation transform method along with Shehu transformation. Laplace and Sumudu transformation have been refined to form Shehu transformation. An algorithm is established for expressing the Shehu transform for the fractional operators like Riemann-Liouville and Caputo by using this new integral transform. Higher-order fractional differential equations are solved in the Caputo sense. Shehu transformation is used to simplify the problems before implementing the decomposition and variational iteration methods to achieve the problem’s comprehensive solutions. This method provides a series form solution with easily computed components and a higher rate of convergence to the exact solution of the targeted problem. The reliability of this process is demonstrated through physical problems. MATLAB software is used to analyze the problems graphically. It is observed that integer-order differential equations do not properly model various phenomena in different fields of science and engineering in relation to fractional-order differential equations. This method is simple and accurate analytical technique that can solve other partial differential equations of fractional order as well.