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This paper deals with the application of non-singular fractional operator in the bioconvection flow of a MHD viscous fluid for vertical surface. The Laplace transform method is used for dimensionless governing equations of momentum, energy and diffusion respectively. Classical governing model is extended to non-integer order approach with non-singular kernel which can be used to describe the memory for natural phenomena. The main advantage is to use this fractional operator can it measure the rate of change at all points of the considered interval, therefore, the present fractional operator incorporate the previous history/memory effects of any system. For the prediction of physical behavior of embedded parameters, some graphs are presented in the graphical section. At the end some remarkable results are found. It is found that non-singular fractional operator measures the memory better in comparison with singular fractional operator. Further, on comparison between different kinds of viscous fluid (Water, Air, Kerosene), it is found that temperature and velocity of air is higher than water and kerosene respectively. The results are validated with the recent published work. [ABSTRACT FROM AUTHOR]

• Fractional nanofluid's model is developed by Caputo time derivative approach. • Enhancement of heat transfer features of nanofluids have made possible the new growth in heat freights and heat fluxes induced by numerous power in miniaturized products, advanced diligences such as in microelectronics, conveyance, fabricating, metrology and defensive structure face cooling systems etc. • Developing heat and mass transfer model of nanofluids by applying fractional calculus enables us to enhance heat and mass transfer phenomenon. • Analytical results are obtained by applying Laplace transform method. • Water is taken as base fluid with five different nano particles and comparison is drawn between them. We have concluded that Ag - water nanofluid has greater temperature field whereas Al 2 O 3 -water has greater velocity as compared to TiO 2 , Cu, Ag, CuO. The present study is aimed to provide the unsteady MHD nanofluid's flow passing through an accelerating infinite vertical plate situated in porous medium. The flow is effected by thermal radiation, Newtonian heating and chemical reaction. Water is considered as conventional base fluid comprising of five different types of nano particles such as Titanium oxide (TiO 2), Aluminium Oxide (Al 2 O 3), Copper Oxide (CuO), Silver (Ag) and Copper (Cu). By using dimensional analysis, the governing equations for temperature, velocity and concentration are reduced to dimensionless and after that these classical equations of present model are generalized to Caputo and Caputo-Fabrizio fractional derivatives. Semi-exact solutions for these equations are obtained via Laplace transform method. Inversion algorithms (Tzou's and Stehfest's) are applied to find the inverse Laplace transform. At last the comparison of water based nanofluids suspended with five different types of nano particles is drawn and effect of nanoparticles as well as fractional parameters (α, β, γ) on temperature and velocity can be seen by software Mathcad. We concluded that Ag -water nanofluid has greater temperature due to its greater value of thermal conductivity as compare to others. Whereas Al 2 O 3 -water has greater velocity because these particles are less denser than TiO 2 , Cu, Ag, CuO. Further we can see that by increasing the value of fractional parameters velocity as well as temperature decreases. Fluid flow can be enhanced with Caputo fractional model while Caputo-Fabrizio decays faster than Caputo and hence well suited in exhibiting the memory of the flow problem at certain time. [ABSTRACT FROM AUTHOR]

Mathematical models are enormously valuable in recognition the characteristics of infectious afflictions. The present study describes and analyses a nonlinear Susceptible-Infected (S·I) type mathematical model for HIV/AIDS. To better comprehend the dynamics of disease diffusion, it is assumed that by giving AIDS patients timely Anti Retroviral Therapy (ART), their transition into HIV infected class is attainable. The ART treatment can reduce or manage the spread of disease among individuals that can extend their life for some more years. For the model, the basic reproduction number is formed which provides a base to study the stability of disease free and endemic equilibria. To understand the entire dynamical behavior of the model, standard finite difference (SFD) schemes such as Runge-Kutta of order four (RK-4) and forward Euler schemes and nonstandard finite difference (NSFD) scheme are implemented. The goal of constructing the NSFD scheme for differential equations is to ensure that it is dynamically reliable, while maintaining important dynamical properties like the positivity of the solutions and its convergence to equilibria of continuous model for all finite step sizes. However, the essential characteristics of the continuous model cannot be properly maintained by the Euler and RK-4 schemes, leading to the possibility of numerical solutions that are not entirely similar to those of the original model. For the NSFD scheme, the Routh-Hurwitz criterion is used to assess the local stability of disease-free and endemic equilibria. To explain the global stability of both the equilibria, Lyapunov functions are offered. To verify the theoretical findings and validate the dynamical aspects of the abovementioned schemes, numerical simulations are also provided. The outcomes offered in this study may be engaged as an effective tool for forecasting the progression of HIV/AIDS epidemic diseases. • A nonlinear Susceptible-Infected (SI) type mathematical model for HIV/AIDS is presented and examined. • The standard finite difference (SFD) schemes, which include forward Euler scheme and Runge-Kutta of order four (RK-4) scheme as well as nonstandard finite difference (NSFD) scheme are designed to understand the entire dynamical behavior of the model. • The Routh-Hurwitz criterion is used to assess the local stability of disease-free and endemic equilibria. • The Lyapunov functions are provided to discuss the global stability of both the equilibria. • To validate the theoretical findings and verify the dynamical characteristics of the aforementioned schemes, numerical simulations are provided. [ABSTRACT FROM AUTHOR]