Recently, novel coronavirus is a serious global issue and having a negative impact on the economy of the whole world. Like other countries, it also effected the economy and people of Pakistan. According to the publicly reported data, the first case of novel corona virus in Pakistan was reported on 27th February 2020. The aim of the present study is to describe the mathematical model and dynamics of COVID-19 in Pakistan. To investigate the spread of coronavirus in Pakistan, we develop the SEIR time fractional model with newly, developed fractional operator of Atangana–Baleanu. We present briefly the analysis of the given model and discuss its applications using world health organization (WHO) reported data for Pakistan. We consider the available infection cases from 19th March 2020, till 31st March 2020 and accordingly, various parameters are fitted or estimated. It is worth noting that we have calculated the basic reproduction number $${\mathfrak{R}}_{0} \approx 2.30748$$R 0 ≈ 2.30748 which shows that virus is spreading rapidly. Furthermore, stability analysis of the model at disease free equilibrium DFE and endemic equilibriums EE is performed to observe the dynamics and transmission of the model. Finally, the AB fractional model is solved numerically. To show the effect of the various embedded parameters like fractional parameter $$\alpha$$α on the model, various graphs are plotted. It is worth noting that the base of our investigation, we have predicted the spread of disease for next 200 days.

The aim of this study is to obtain the closed form solutions for the laminar and unsteady couple stress fluid flow ℓ. The fluid is allowed to flow between two infinite parallel plates separated by distance. Moreover, we have considered that the lower plate is moving with uniform velocity 0 U and upper plate is stationary. For this purpose, engine oil is taken as a base fluid and to enhance the efficiency of lubricating oil, Molybdenum disulphide nanoparticles are dispersed uniformly in the engine oil. The flow is formulated mathematically in terms of partial differential equations of order four. Furthermore, the derived system of partial differential equations are fractionalized by using the mostly used definition of Caputo-Fabrizio time fractional derivative. The more general exact solutions for velocity, temperature and concentration distributions are obtained by using the joint applications of Fourier and the Laplace transforms. The effect of different parameters of interest of the obtained general solutions are discussed by sketching graphs. Furthermore, substituting favorable limits of different parameters, four different limiting cases are recovered from our obtained general solutions i.e. (a) Couette flow (b) Classical couple stress fluid (c) Newtonian viscous fluid and (d) in the absence of thermal and concentration. Moreover, the effect of different physical parameters on the velocity, temperature and concentration distributions are discussed graphically. It is worth noting that couple stress parameter corresponds to a decrease in the velocity profile. In order to observe the differences clearly, all the figures are compared for integer order and fractional order which provide a more realistic approach as compared to the classical model. Additionally, skin friction is calculated at lower as well as upper plate. Nusselt number and Sherwood number are also tabulated. It is noticed that the rate of heat transfer of engine oil can be enhanced up to 12.38% and decrease in mass transfer up to 2.14% by adding Molybdenum disulphide nanoparticles in regular engine oil.

Several scientists are interested in recent developments in nanotechnology and nanoscience. Grease is an essential component of many machines and engines because it helps keep them cool by reducing friction between their various elements. In sealed life applications including centralized lubrication systems, electrical motors, bearings, logging and mining machinery, truck wheel hubs, construction, landscaping, and gearboxes, greases are also utilized. Nanoparticles are added to convectional grease to improve its cooling and lubricating properties. More specifically, the current study goal is to investigate open channel flow while taking grease into account as a Maxwell fluid with MoS₂ nanoparticles suspended in it. The Caputo-Fabrizio time-fractional derivative is used to convert the issue from a linked classical order PDE to a local fractional model. To determine the precise solutions for the velocity, temperature, and concentration distributions, two integral transform techniques the finite Fourier sine and the Laplace transform technique are jointly utilized. The resultant answers are physically explored and displayed using various graphs. It is important to note that the fractional model, which offers a variety of integral curves, more accurately depicts the flow behavior than the classical model. Skin friction, the Nusselt number, and the Sherwood number are engineering-related numbers that are quantitatively determined and displayed in tabular form. It is determined that adding MoS₂ nanoparticles to grease causes a 19.1146% increase in heat transmission and a 2.5122% decrease in mass transfer. The results obtained in this work are compared with published literature for the accuracy purpose.

Gold nanoparticles are commonly used as a tracer in laboratories. They are biocompatible and can transport heat energy to tumor cells via a variety of clinical techniques. As cancer cells are tiny, properly sized nanoparticles were introduced into the circulation for invasion. As a result, gold nanoparticles are highly effective. Therefore, the current research investigates the magnetohydrodynamic free convection flow of Casson nanofluid in an inclined channel. The blood is considered as a base fluid, and gold nanoparticles are assumed to be uniformly dispersed in it. The above flow regime is formulated in terms of partial differential equations. The system of derived equations with imposed boundary conditions is non-dimensionalized using appropriate dimensionless variables. Fourier's and Fick's laws are used to fractionalize the classical dimensionless model. The Laplace and Fourier sine transformations with a new transformation are used for the closed-form solutions of the considered problem. Finally, the results are expressed in terms of a specific function known as the Mittag-Leffler function. Various figures and tables present the effect of various physical parameters on the achieved results. Graphical results conclude that the fractional Casson fluid model described a more realistic aspect of the fluid velocity profile, temperature, and concentration profile than the classical Casson fluid model. The heat transfer rate and Sherwood number are calculated and presented in tabular form. It is worth noting that increasing the volume percentage of gold nanoparticles from 0 to 0.04 percent resulted in an increase of up to 3.825% in the heat transfer rate.

fferent from a Newtonian fluid, Couple stress fluid (CSF), includes a new material constant which is responsible for couple stress and the lubricant viscosity. This material constant comes with the 4th order spatial derivative term, and due to this higher order derivative term in the momentum equation, this fluid (CSF) is comparatively less investigated even for the classical fluid problems. This article aims to study the fractional model of CSF, based on Atangana–Baleanu (AB) fractional derivatives definition. Since this AB definition is new, therefore, for the sake of comparison and correctness, this problem is also solved using Caputo–Fabrizio (CF) fractional derivative definition. CSF is considered to flow between two parallel plates one of which is at rest and the other is moving with constant velocity. The external pressure gradient is also applied. This type of flow situations is usually called as generalized Couette flow. The problem is first written in dimensionless form and then solved for the exact solution using the Laplace transform and finite Fourier sine transform. CSF velocity obtained via AB fractional derivative is compared with CSF velocity obtained via CF fractional derivative approach and the results obtained are shown graphically. CSF results for interesting fluid parameters are displayed in various graphs for both AB and CF fractional derivatives. It is observed that the CSF velocities obtained with AB and CF fractional derivatives are the same for unit time. For time less than one and greater than one variation in CSF velocities is observed. In limiting sense, the present CSF solutions are reduced to a similar Newtonian fluid problem solution in the absence of external pressure gradient.