The current paper examines some novel and interesting aspects of the fractional biological population model involving the efficacious Atangana-Baleanu fractional derivative operator. It assists us in comprehending the dynamical techniques of population changes in biological population models and generates precise prognostication. This technique correlates with the Elzaki transform method and the homotopy perturbation method. The Elzaki transform is a modification of the classical Fourier Laplace transform. The approximate-analytical solutions of the biological population model are examined using the Elzaki transform homotopy perturbation method ( ETHPM ). The exact solution of the aforesaid scheme is being investigated in terms of the Mittag-Leffler function. The role of fractional-order on spatial diffusion of a biological population model is demonstrated in two and three-dimensional surface plots. The comparative analysis between exact and numerical solutions reveals the innovative features of the composite fractional derivative in the discussed model. Furthermore, the proposed approach is very powerful, reliable, well-organized, and pragmatic for fractional PDEs and it might be extended to other physical processes.

This study aims at investigating the numerical analysis of pollutant transport in homogeneous porous media with solid plate stacks. The investigation was performed for solid/impervious objects of the same size placed in homogeneous porous media. The pollutant transport equation (i.e., steady-state and time dependent advection-dispersion) chosen in mathematical modeling. Three cases arise on the basis of dispersion coefficients: (a) when dispersion is uniformly constant, (b) when dispersion depends upon magnitude of the velocity, and (c) when dispersion depends upon magnitude of the velocity and directional dispersivities, all these are discussed in detail. Generally, 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, one-sided, backward and forward finite difference formulae of the same order are used to discretize the domain. Simulations of velocity potential and stream function are approximated by Matlab software. Then equipotential lines and streamlines are visualized in the form of contours. Both, velocity potential and stream function are harmonic and satisfy Laplace's equation. Fluid flow lines and pollutant concentration are represented graphically for several parameters involved in the study. It is found that entrance/exit length, shape, hydraulic conductivity, the number and position of impervious objects affect the fluid flow and pollutant transport. However, there is no significant affect of heated objects on pollutant transport. Moreover, advection and dispersion depend upon permeability of porous media and properties of solid matrix. To authenticate the Matlab scheme of finite difference, it is verified that fluid as well as pollutant fluxes (in and out) are equal. Moreover, time-dependent problem converges to steady-state form after very long time. For monitoring or forecasting the build up of contamination concentration, the pollutant transport model is considerable. As this model is affected by different parameters which are discussed above, can helps to overcome the pollutant accumulation. The solid object is main key to lessen the contamination in the underground. If the entrance or leakage point of the domain is blocked by impermeable object or filled the vertical column with material of low hydraulic conductivity it ultimately slows down or even refrains the pollutant particles to pass through. The pollutant concentration is also minimized by injecting the bioremedial agents with the help of treatment columns.

The current paper examines some novel and interesting aspects of the mathematical modelling of fluid flow and forced convection heat transfer from a plate stack embedded in a homogeneous porous media. Time-dependent forced convection heat transfer from some cylinders embedded in groundwater are explored numerically using finite difference methods. An algorithm was developed about rectangular heated solid objects with the same size placed in an in-line arrangement in homogeneous porous media. Central and one-sided finite difference formulae are used to discretize the domain. Applying the mathematical model, continuity, momentum, and energy equations are solved by the finite difference methods. Generally, analytical solutions are not possible for such problems, so most of the work is done numerically. The numerical results for the two-dimensional velocity potential and stream function have been calculated and the resulting fluid and temperature contours are represented graphically for various values of the parameters involved (e.g. length of the entrance, the fluid flow rates, coefficient of thermal diffusion and advection in the observed long-time plume behaviour, effect of time and temperature on heat transfer, etc.). It is observed that fluid and heat fluxes and fluid velocity depend upon the size, shape, burial depth, position of heated objects, and positions of the entrance and exit of the homogeneous porous media. Moreover, the hydraulic conductivity of the porous medium affects the pressure drops, fluid velocity as well as the heat transfer rates.