In this study, we propose new illustrative and effective modeling to point out the behaviors of the Hepatitis-B virus (Hepatitis-B). Not only do we consider the mathematical modeling, equilibria, stabilities, and existence–uniqueness analysis of the model, but also, we make numerical simulations by using the Adams–Bashforth numerical scheme. However, we apply the parameter estimation method to determine our model parameters and find the curve that best fits the model. Additionally, in this study, the stability analysis of the aforementioned model is considered, and also the sensitivity analysis of R₀ is examined. The results point out that the order of the fractional derivative has an essential effect on the dynamical process of the constructed model for Hepatitis-B.

The Navier–Stokes (NS) equations involving MHD effects with time-fractional derivatives are discussed in this paper. This paper investigates the local and global existence and uniqueness of the mild solution to the NS equations for the time fractional differential operator. In addition, we work on the regularity effects of such types of equations which are caused by MHD flow.

In recent times, all world banks have been threatened by the liquidity risk problem. This phenomenon represents a devastating financial threat to banks and may lead to irrecoverable consequences in case of negligence or underestimation. In this article, we study a mathematical model that describes the contagion of liquidity risk in the banking system based on the SIR epidemic model simulation. The model consists of three ordinary differential equations illustrating the interaction between banks susceptible or affected by liquidity risk and tending towards bankruptcy. We have demonstrated the bornness and positivity of the solutions, and we have mathematically analyzed this system to demonstrate how to control the banking system’s stability. Numerical simulations have been illustrated by using real data to support the analytical results and prove the effects of different system parameters studied on the contagion of liquidity risk.

This work proposes a qualitative study for the fractional second-grade fluid described by a fractional operator. The classical Caputo fractional operator is used in the investigations. The exact analytical solutions of the constructed problems for the proposed model are determined by using the Laplace transform method, which particularly includes the Laplace transform of the Caputo derivative. The impact of the used fractional operator is presented; especially, the acceleration effect is noticed in the paper. The parameters’ influences are focused on the dynamics such as the Prandtl number $\left(Pr\right)$, the Grashof numbers $\left(Gr\right)$, and the parameter $\eta$ when the fractional-order derivative is used in modeling the second-grade fluid model. Their impacts are also analyzed from a physical point of view besides mathematical calculations. The impact of the fractional parameter $\alpha$ is also provided. Finally, it is concluded that the graphical representations support the theoretical observations of the paper.

In this article, unsteady free convective heat transport of copper-water nanofluid within a square-shaped enclosure with the dominance of non-uniform horizontal periodic magnetic effect is investigated numerically. Various nanofluids are also used to investigate temperature performance. The Brownian movement of nano-sized particles is included in the present model. A sinusoidal function of the y coordinate is considered for the magnetic effect, which works as a non-uniform magnetic field. The left sidewall is warmed at a higher heat, whereas the right sidewall is cooled at a lower heat. The upper and bottom walls are insulated. For solving the governing non-linear partial differential equation, Galerkin weighted residual finite element method is devoted. Comparisons are made with previously published articles, and we found there to be excellent compliance. The influence of various physical parameters, namely, the volume fraction of nanoparticles, period of the non-uniform magnetic field, Rayleigh number, the shape and diameter of nanoparticles, and Hartmann number on the temperature transport and fluid flow are researched. The local and average Nusselt number is also calculated to investigate the impact of different parameters on the flow field. The results show the best performance of heat transport for the Fe₃O₄-water nanofluid than for other types of nanofluids. The heat transport rate increases 20.14% for Fe₃O₄-water nanofluid and 8.94% for TiO₂-water nanofluid with 1% nanoparticles volume. The heat transportation rate enhances with additional nanoparticles into the base fluid whereas it decreases with the increase of Hartmann number and diameter of particles. A comparison study of uniform and non-uniform magnetic effects is performed, and a higher heat transfer rate is observed for a non-uniform magnetic effect compared to a uniform magnetic effect. Moreover, periods of magnetic effect and a nanoparticle’s Brownian movement significantly impacts the temperature transport and fluid flow. The solution reaches unsteady state to steady state within a very short time.