In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number ${\mathcal{R}}_0$ was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

This study examines the bidimensional nonlinear convective flow of ternary hybrid nanofluid upon a nonlinear stretching sheet. Three types of nanoparticles, namely , are suspended in the base fluid taken as water with a new composition which is termed as ternary hybrid nanofluid. To stabilize the flow and thermal properties of the new composition, the Brownian as well as thermophoresis properties are incorporated into energy and mass equations. The nonlinear thermal radiations and heat absorption/generation terms are included in the energy equation. The effects of the Darcy–Forchheimer phenomenon are also induced in the momentum equation. The set of model equations has shifted to dimension-free form by employing suitable variables. It has concluded in this study that flow characteristics have been declined with augmenting values of volumetric fractions of solid nanoparticles, porosity, and inertia factors and have upsurge with higher values of thermal and nonlinear thermal Grashof numbers. Thermal characteristics have been observed to be augmented with growth in radiation, Brownian motion, thermophoresis, heat generation/absorption, temperature ratio factors, and volumetric fraction of solid nanoparticles. These effects are more significant for ternary hybrid nanoparticles. Concentration profiles have been declined with higher values of Brownian motion factor, Lewis number, and upsurge with growth in thermophoresis factor. It has also been deduced in this investigation that the thermal flow rate is higher for trihybrid nanofluid than hybrid or traditional nanofluids, and a percentage growth in Nusselt number has been shown through statistical chart in support of this work. Current results have been compared with established results and found a fine agreement amongst all results.

This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the $\rho$-Laplace homotopy transform method ($\rho$-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders $\rho$ and $\phi$.