It is well known that the solution of fractional models proved to be a powerful tool in studying various problems which appear in the sciences of real life. In view of the fact that economic applications are accelerating at an amazing pace, and the large number of modeling in this speciality, it has expanded the number of problems. So, our contribution is based on finding generalized solutions of a fractional differential equation known for their applications in microeconomics and finance and creating an algorithm which allows us to estimate the coefficients of this type of equation. And to really illustrate our results we will choose a model known in the stochastic literature by COGARCH but with fractional derivative, to demonstrate the asymptotic behavior of the estimators, including the impact of fractional order on the space of stochastic differential equations.

This paper analyzes the effects of yield stress and nanoparticles transport on the natural convection of viscoplastic Casson nanofluids. The non-linear coupled partial differential equations are solved numerically using Buongiorno’s mathematical model. The governing parameters for the problem are the Rayleigh number (Ra), yield number (Y), and thermophoresis and Brownianmotion parameters (N t&Nb). The effects of these parameters on the fluid flow, heat and mass transfer, and the shape of yielded and unyielded regions are examined and discussed in detail. The results demonstrate that the heat and mass transfer rates increase as the Rayleigh number increases, while the opposite behaviors are observed with increasing the yield number. The fluid is difficultly yielded at low Rayleigh number. The heat and mass transfer are primarily due to conduction at the high values of the yield number. The main effect of thermophoresis and Brownian motion parameters is on temperature and concentration distribution in the cavity. These parameters also show significant impacts on critical heat and mass transfer.

In this paper, Brownian motion of nanoparticles and clusters and resulted micromixing are combined with the aggregation kinetics of clusters to capture the effects of nanoparticles on the thermal conductivity of nanofluids. Starting from kinetic theory, random motion of nanoparticles and induced micro-convection were combined with the kinetics of aggregates and a theoretical model which depends on two semi-empirical parameters was derived. The proposed model attributes the thermal conductivity of nanofluids not only to the intrinsic physical properties, but also to physicochemical parameters which affect the stability state of nanofluids. The more nanofluid is stabilized, the more keff increases. We have also demonstrated that the thermal conductivity either increases or decreases with respect to the particle size and there is an optimum value for the particle radius at which the thermal conductivity of the nanofluid is maximum. Depending on the chemistry of the solution, the optimized radius of the nanoparticles in a suspension depends on the temperature and pH of the suspension and the volume fraction of the nanoparticles. This behaviour is not feasible without including the effects of aggregation kinetics combined with Brownian motion and induced micro-convection.

Finite element method (FEM) is used to study the hydrothermal characteristics of the nano-fluid subjected to Brownian motion. For effective thermal conductivity and effective, viscosity Koo-Kleinstreuer-Li (KKL) model is used. It is observed that the dispersion of nano-particles in Newtonian liquid causes a significant increase in the effective thermal conductivity. This results based on the dispersion of nano-particles help engineers to design an efficient thermal system. A significant role of viscous dissipation on diffusion of momentum of wall into the fluid is observed. Therefore, dissipations effects cannot be ignored while designing thermal systems. The buoyant force is responsible for the effect of electromagnetic thermal radiations on the velocity of fluid convectively heated surface enhances the rate of generation of entropy. This study also recommends that nano-fluids are the best coolants as compare to the base fluids. Imposition of magnetic field causes more entropy generation.