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This article is mainly concerned to link the Hermite-Hadamard and the Jensen-Mercer inequalities by using majorization theory and fractional calculus. We derive the Hermite-Hadamard-Jensen-Mercer-type inequalities in conticrete form, which serve as both discrete and continuous inequalities at the same time, for majorized tuples in the framework of the famous Atangana-Baleanu fractional operators. Also, the main inequalities encompass the previously established inequalities as special cases. Using generalized Mercer’s inequality, we also investigate the weighted forms of our major inequalities for certain monotonic tuples. Furthermore, the derivation of new integral identities enables us to construct bounds for the discrepancy of the terms concerning the main results. These bounds are constructed by incorporating the convexity of ∣f′∣| f^{\prime} | and ∣f′∣q(q>1){| f^{\prime} | }^{q}\hspace{0.33em}\left(q\gt 1) and making use of power mean and Hölder inequalities along with the established identities.

Abstract SARS‐CoV‐2 epidemic (severe acute respiratory corona virus 2 syndromes) has caused major impacts on a global scale. Several countries, including India, Europe, U.S.A., introduced a full state/nation lockdown to minimise the disease transmission through human interaction after the virus entered the population and to minimise the loss of human life. Millions of people have gone unemployed due to lockdown implementation, resulting in business and industry closure and leading to a national economic slowdown. Therefore, preventing the spread of the COVID‐19 virus in the world while also preserving the global economy is an essential problem requiring an effective and immediate solution. Using the compartmental epidemiology S, E, I, R or D (Susceptible, Exposed, Infectious, Recovery or Death) model extended to multiple population regions, the authors predict the evolution of the SARS‐CoV‐2 disease and construct an optimally scheduled lockdown calendar to execute lockdown over phases, using the well‐known Knapsack problem. A comparative analysis of both classical and quantum models shows that our model decreases SARS‐CoV‐2 active cases while retaining the average global economic factor, Gross Domestic Product, in contrast to the scenario with no lockdown.

The purpose of this research elaborates the behaviour of the two-phase hybrid nano-fluid flow of viscous fluid through the stretching sheet. Impacts of heat transfer, for MHD two-dimensional steady flow over stretching surface with Cattaneo-Christov (CC) model are discussed. Bio-convective, thermal radiation and convective conditions are considered. Porous surface and velocity slip effects are also considered. Darcy-Forchheimer relation characterizes porous medium. The governing equations are solved through MATLAB. Plots of temperature and velocity fields are discussed via various estimations of emerging variables. Results findings shows the velocity profile decreases in case of both simple and hybrid nano-fluids for higher estimations of magnetic parameter and local inertia coefficient parameter. Fluid temperature reduces for maximum thermal relaxation parameter and possess higher values against radiation parameter. Microorganism profile reduces bio-convection Lewis number but increases for the magnetic parameter. Prominent results from this study include determining the effective parameters to enhance heat transfer efficiency and defining the circumstances in which the nano-fluid performs better than traditional fluids. These discoveries offer significant perspectives on the utilization of Cattaneo-Christov model with hybrid nano-fluid flow in comprehensive thermal management systems and establish a basis for subsequent practical and theoretical investigations within the domain.

The complex phenomenon of heat transfer plays a pivotal role in a broad spectrum of mechanical and industrial applications, ranging from space and nuclear reactor cooling to medicinal applications like magnetic drug targeting, energy production, heat conduction in biological tissues, among others. Historically, Fourier's law of heat conduction has served as a predictive model for heat transfer in numerous practical scenarios. However, this model's limitation is its generation of a parabolic energy expression, implying an immediate influence on the system by any initial disturbance. Numerous researchers have attempted to address this issue, leading to the Cattaneo's modification to Fourier's law, which introduces a relaxation time for heat flux, meaning the time needed to achieve stable heat conduction following the imposition of a temperature gradient. Christov later tweaked this model into a material invariant version, integrating the upper-connected derivative from the Oldroyd model. These models, now collectively termed as the Cattaneo-Christov (CC) model, form the core of the present study. In this study, we explore the mixed convective MHD Falkner-Skan Suttberby nanofluid flow towards a wedge surface, in the presence of a mixed convection. A two-phase nanofluid based C–C model is utilized to apply the nano-concept. The nonlinear equation system is solved using the bvp4c technique in MATLAB. The effects of various parameters including the magnetic field strength, wedge angle, nanoparticle volume fraction, and Prandtl number on the velocity and temperature profiles are discussed. The results show that the velocity increases with the magnetic parameter, wedge angle, and nanoparticle fraction, while the temperature decreases or increases depending on the suction or injection conditions. The Nusselt numbers and solutal transport rate are also reported for different parameter values. Overall, the study demonstrates that the non-Fourier C–C model yields significantly different solutal and heat transfer characteristics compared to the classical Fourier law. The results provide useful insights into the heat transfer behavior of nanofluids in stagnation point flows over geometries with complex shapes, which have practical applications in areas such as aerodynamics and thermal management of electronic devices.

The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣f′∣| f^{\prime} | and ∣f′∣q(q>1)| f^{\prime} {| }^{q}\hspace{0.33em}\left(q\gt 1) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.

Influences of Soret (thermo-diffusion) and Dufour (diffusion-thermo) effects for MHD second grade nanofluid flow by a stretchable Riga surface is investigated. The Cattaneo-Christove (CC) method is used to predict heat and mass transport. Thermal radiation and thermal solutal stratification effects are taken into account. Heat generation, mixed convection, thermophoresis and Brownian motion effects are included. Temperature-dependent thermal conductivity and concentration-dependent mass diffusivity are adopted. Formulation for non-similar analysis is organized. Optimal homotopy analysis (OHAM) approach is used to compute solutions of resulting problems. Average square residual error (TSRE) is calculated. Outcomes of influential variables on interesting quantities are outlined. Velocity enhances for higher values of buoyancy parameter and mixed convection parameters. Temperature decays is noted for stratification parameter but opposite result observed for Dufour number.

Education is a purposeful activity and has great implications for both the individuals and the societies. This paper examines the social and economic benefits of education in developing countries in general and Pakistan as a developing country, in particular. The study is ex-post facto research in nature and primarily based on the secondary data. The study explores that the monetary and non-monetary benefits of education to the society are numerous such employment, earning, poverty alleviation, export rate. On economic front while social responsibility, health care, political awareness, etc on the social front. The government and the civil society have to invest in education to reap the benefits of education more socially than economically. This is the only way to make Pakistan a peaceful and safer place rather through other measures. The security insurance consequently, affects the country's economy positively.

The main of this article is to analyze magnetohydrodynamic bioconvective flow of Sutterby nanoliquid. Gyrotactic microorganism in presence of chemical reaction is addressed. Thermophoretic, magnetic field, random motion heat generation and radiation are discussed. Furthermore, Dufour and Soret behaviors are taken into account. Thermal conduction augmentation performance is discussed by utilization Boungiorno's model. Nonlinear PDE's (partial differential equations) are changed to ordinary system through appropriate variables. To developed computational solutions, we used the ND-solve technique. Results for temperature, microorganism field, liquid flow, and concentration are exhibited through different emerging variables. The physical quantities like Nusselt number, microorganism density number and solutal transport rate for various sundry variables are presented. Summary of main results re highlighted in the conclusions. Velocity reduces against magnetic field, while reverse trend seen for buoyancy ratio variable. Thermal distribution has an enhancing trend for magnetic and radiation variables. An enhancement in concentration distribution is seen for Soret number.