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In this article, we investigate the behavior of weak solutions for the three-dimensional Navier-Stokes-Voigt-Brinkman-Forchheimer fluid model with memory and Tresca friction law within a thin domain. We analyze the asymptotic behavior as one dimension of the fluid domain approaches zero. We derive the limit problem and obtain the specific Reynolds equation, while also establishing the uniqueness of the limit velocity and pressure distributions.

Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number (R0)({R}_{0}). Both disease-free equilibrium (DFE) (E0)({E}^{0}) and the endemic equilibrium (E⁎)({E}^{\ast }) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that (E0)({E}^{0}) is locally asymptotically stable when R01{R}_{0}\gt 1. We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium \text{ }and DFE based on the number (R0)({R}_{0}). To ascertain the dominance of the parameters, we examined the sensitivity of the number (R0)({R}_{0}) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

In previous work, Fayssal considered a thermoelastic laminated beam with structural damping, where the heat conduction is given by the classical Fourier’s law and acting on both the rotation angle and the transverse displacements established an exponential stability result for the considered problem in case of equal wave speeds and a polynomial stability for the opposite case. This article deals with a laminated beam system along with structural damping, past history, and the presence of both temperatures and microtemperature effects. Employing the semigroup approach, we establish the existence and uniqueness of the solution. With the help of convenient assumptions on the kernel, we demonstrate a general decay result for the solution of the considered system, with no constraints regarding the speeds of wave propagations. The result obtained is new and substantially improves earlier results in the literature.

Ultrasound imaging stands as a cornerstone of modern medical diagnostics, revolutionising clinical practice with its non-invasive, real-time visualisation of internal structures. Central to this technique is the propagation of ultrasound waves and their intricate interplay with biological tissues, culminating in the generation of intricate and detailed images. This study delves into the symbiotic relationship between solitons and ultrasound imaging within the framework of the Westervelt equation, a fundamental model governing ultrasound propagation. Employing the generalised Riccati equation mapping method and the generalised exponential rational function method, a diverse array of soliton solutions is elucidated, encompassing dark, kink, combined dark–bright, combined dark-singular, periodic singular, and singular solitons. Visualisation of these solutions through 3D plots, contour plots, and 2D plots at varying time intervals offers a captivating insight into their dynamic nature. We provide a comparison of these solutions through 2D plots at different parameter values, highlighting their varying impacts. Central to this study is the exploration of how these soliton solutions can be harnessed to enhance the quality and accuracy of ultrasound images in medical imaging. Through meticulous analysis of their characteristics, this research seeks to illuminate their potential applications, paving the way for a new era of precision diagnostics in healthcare. By conducting thorough mathematical analyses and numerical simulations, we seek to elucidate the complex relationship between soliton theory and ultrasound imaging, connecting the theoretical aspects of nonlinear wave phenomena with their practical applications in medical diagnostics. An intensive literature review underscores the novelty of our work.