This article investigates natural convection with double-diffusive properties numerically in a vertical bi-layered square enclosure. The cavity has two parts: one part is an isotropic and homogeneous porous along the wall, and an adjacent part is an aqueous fluid. Adiabatic, impermeable horizontal walls and constant and uniform temperatures and concentrations on other walls are maintained. To solve the governing equations, the finite element method (FEM) employed and predicted results shows the impact of typical elements of convection on double diffusion, namely the porosity thickness, cavity rotation angle, and thermal conductivity ratio. Different Darcy and Rayleigh numbers effects on heat transfer conditions were investigated, and the Nusselt number in the border of two layers was obtained. The expected results, presented as temperature field (isothermal lines) and velocity behavior in X and Y directions, show the different effects of the aforementioned parameters on double diffusion convective heat transfer. Also results show that with the increase in the thickness of the porous layer, the Nusselt number decreases, but at a thickness higher than 0.8, we will see an increase in the Nusselt number. Increasing the thermal conductivity ratio in values less than one leads to a decrease in the average Nusselt number, and by increasing that parameter from 1 to 10, the Nusselt values increase. A higher rotational angle of the cavity reduces the thermosolutal convective heat transfer, and increasing the Rayleigh and Darcy numbers, increases Nusselt. These results confirm that the findings obtained from the Finite Element Method (FEM), which is the main idea of this research, are in good agreement with previous studies that have been done with other numerical methods.

The influence of boundary layer flow of heat transfer analysis on hybrid nanofluid across an extended cylinder is the main focus of the current research. In addition, the impressions of magnetohydrodynamic, porous medium and thermal radiation are part of this investigation. Arrogate similarity variables are employed to transform the governing modelled partial differential equations into a couple of highly nonlinear ordinary differential equations. A numerical approach based on the BVP Midrich scheme in MAPLE solver is employed for solution of the set of resulting ordinary differential equations and obtained results are compared with existing literature. The effect of active important physical parameters like Magnetic Field, Porosity parameter, Eckert number, Prandtl number and thermal radiation parameters on dimensionless velocity and energy fields are employed via graphs and tables. The velocity profile decreased by about 65% when the magnetic field parameter values increases from 0.5 to 1.5. On the other hand increased by 70% on energy profile. The energy profile enhanced by about 62% when the Radiation parameter values increases from 1.0 < Rd < 3.0. The current model may be applicable in real life practical implications of employing Engine oil-SWCNTs-MWCNTs-TiO₂ nanofluids on cylinders encompass enhanced heat transfer efficiency, and extended component lifespan, energy savings, and environmental benefits. This kind of theoretical analysis may be used in daily life applications, such as engineering and automobile industries.

In this article, examine the performance of a physics informed neural networks (PINN) intelligent approach for predicting the solution of non-linear Lorenz differential equations. The main focus resides in the realm of leveraging unsupervised machine learning for the prediction of the Lorenz differential equation associated particle swarm optimization (PSO) hybridization with the neural networks algorithm (NNA) as ANN-PSO-NNA. In particular embark on a comprehensive comparative analysis employing the Lorenz differential equation for proposed approach as test case. The nonlinear Lorenz differential equations stand as a quintessential chaotic system, widely utilized in scientific investigations and behavior of dynamics system. The validation of physics informed neural network (PINN) methodology expands to via multiple independent runs, allowing evaluating the performance of the proposed ANN-PSO-NNA algorithms. Additionally, explore into a comprehensive statistical analysis inclusive metrics including minimum (min), maximum (max), average, standard deviation (S.D) values, and mean squared error (MSE). This evaluation provides found observation into the adeptness of proposed AN-PSO-NNA hybridization approach across multiple runs, ultimately improving the understanding of its utility and efficiency.

To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases which are spread in the world wide. The objective of the research study is to assess the early diagnosis and treatment of cholera virus by implementing remedial methods with and without the use of drugs. A mathematical model is built with the hypothesis of strengthening the immune system, and a ABC operator is employed to turn the model into a fractional-order model. A newly developed system SEIBR, which is examined both qualitatively and quantitatively to determine its stable position as well as the verification of flip bifurcation has been made for developed system. The local stability of this model has been explored concerning limited observations, a fundamental aspect of epidemic models. We have derived the reproductive number using next generation method, denoted as “ $$R_{0}$$R 0 ”, to analyze its impact rate across various sub-compartments, which serves as a critical determinant of its community-wide transmission rate. The sensitivity analysis has been verified according to its each parameters to identify that how much rate of change of parameters are sensitive. Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values which is advanced tool for reliable bounded solution. Also the error analysis has been made for developed scheme. Simulations have been made to see the real behavior and effects of cholera disease with early detection and treatment by implementing remedial methods without the use of drugs in the community. Also identify the real situation the spread of cholera disease after implementing remedial methods with and without the use of drugs. Such type of investigation will be useful to investigate the spread of virus as well as helpful in developing control strategies from our justified outcomes.

COVID-19 is linked to diabetes, increasing the likelihood and severity of outcomes due to hyperglycemia, immune system impairment, vascular problems, and comorbidities like hypertension, obesity, and cardiovascular disease, which can lead to catastrophic outcomes. The study presents a novel COVID-19 management approach for diabetic patients using a fractal fractional operator and Mittag-Leffler kernel. It uses the Lipschitz criterion and linear growth to identify the solution singularity and analyzes the global derivative impact, confirming unique solutions and demonstrating the bounded nature of the proposed system. The study examines the impact of COVID-19 on individuals with diabetes, using global stability analysis and quantitative examination of equilibrium states. Sensitivity analysis is conducted using reproductive numbers to determine the disease’s status in society and the impact of control strategies, highlighting the importance of understanding epidemic problems and their properties. This study uses two-step Lagrange polynomial to analyze the impact of the fractional operator on a proposed model. Numerical simulations using MATLAB validate the effects of COVID-19 on diabetic patients and allow predictions based on the established theoretical framework, supporting the theoretical findings. This study will help to observe and understand how COVID-19 affects people with diabetes. This will help with control plans in the future to lessen the effects of COVID-19.

Understanding and controlling the thermal transport phenomena are crucial in numerous applications. The current research emphasizes thermal diffusivity of an inclined magnetized Cross fluid with temperature-dependent thermal conductivity with a computational iterative spectral relaxation scheme. Cross mathematical model is employed to characterizes non-Newtonian behavior and to uncover viscoelastic properties of fluid. Flow is incorporated under temperature thermal influence and external inclined magnetic strength is considered for thermal variations. Various prominent factors, including cross index, magnetic field, inclination angle, temperature-dependent thermal conductivity are analyzed on the fluid's thermal diffusivity. The flow governing PDEs are converted into system of ODEs by using suitable transformation. Spectral relaxation computation scheme is then used for controlling the new set equations. SRM algorithm controlling subsystems is built through MATLAB. Numerical results are illustrated by MATLAB graphs. Physical quantities such as Sherwood numbers, Nusselt and skin friction coefficient are visually taken place through statistical graphs with two cases of imposed magnetic field. The results of this investigation shed light on how non-Newtonian fluids behave when exposed to temperature changes and magnetic fields and useful in understanding and leverage these effects for specific applications.