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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.

In this study, we address a Cauchy problem within the context of the one-dimensional Timoshenko system, incorporating a distributed delay term. The heat conduction aspect is described by the Lord-Shulman theory. Our demonstration establishes that the dissipation resulting from the coupling of the Timoshenko system with Lord-Shulman’s heat conduction is sufficiently robust to stabilize the system, albeit with a gradual decay rate. To support our findings, we convert the system into a first-order form and, utilizing the energy method in Fourier space, and derive point wise estimates for the Fourier transform of the solution. These estimates, in turn, provide evidence for the slow decay of the solution.

Fractional diffusion partial differential equation (PDE) models are used to describe anomalous transport phenomena in fractal porous media, where traditional diffusion models may not be applicable due to the presence of long-range dependencies and non-local behaviors. This study presents an efficient hybrid meshless method to the compute numerical solution of a two-dimensional multiterm time-fractional convection-diffusion equation. The proposed meshless method employs multiquadric-cubic radial basis functions for the spatial derivatives, and the Liouville-Caputo derivative technique is used for the time derivative portion of the model equation. The accuracy of the method is evaluated using error norms, and a comparison is made with the exact solution. The numerical results demonstrate that the suggested approach achieves better accuracy and computationally efficient performance.

Hepatitis B, a liver disease caused by the hepatitis B virus (HBV), poses a significant public health burden. The virus spreads through the exchange of bodily fluids between infected and susceptible individuals. Hepatitis B is a complex health challenge for individuals. In this research, we propose a nonlinear HBV mathematical model comprising seven compartments: susceptible, latent, acutely infected, chronically infected, carrier, recovered, and vaccinated individuals. Our model investigates the dynamics of HBV transmission and the impact of vaccination on disease control. Using the next-generation matrix approach, we derive the basic reproduction number R0{R}_{0} and determine the disease-free equilibrium points. We establish the global and local stability of the model using the Lyapunov function. The model is numerically solved using the higher-order Galerkin time discretization technique, and a comprehensive sensitivity analysis is carried out to investigate the impact of all physical parameters involved in the proposed nonlinear HBV mathematical model. A comparison was made of the accuracy and dependability with the findings produced using the Runge–Kutta fourth-order (RK4) approach. The findings highlight the critical need for vaccination, particularly among the exposed class, to facilitate rapid recovery and mitigate the spread of HBV. The results of this study provide valuable insights for public health policymakers and inform strategies for hepatitis B control and elimination.

The Lord Shulman swelling porous thermo-elastic soil system with the presence of a distributed delay term is studied in this work. We will establish the well-posedness of the system and the exponential stability of the system is derived.

Mathematical models for infectious diseases can help researchers, public health officials, and policymakers to predict the course of an outbreak. We formulate an epidemic model for the transmission dynamics of Zika infection with carriers to understand the intricate progression route of the infection. In our study, we focused on the visualization of the transmission patterns of the Zika with asymptomatic carriers, using fractional calculus. For the validity of the model, we have shown that the solutions of the system are positive and bounded. Moreover, we conduct a qualitative analysis and examine the dynamical behavior of Zika dynamics. The existence and uniqueness of the solution of the system have been proved through analytic skills. We establish the necessary conditions to ensure the stability of the recommended system based on the Ulam–Hyers stability concept (UHS). Our research emphasizes the most critical factors, specifically the mosquito biting rate and the existence of asymptomatic carriers, in increasing the complexity of virus control efforts. Furthermore, we predict that the asymptomatic fraction has the ability to spread the infection to non-infected regions. Furthermore, treatment due to medication, the fractional parameter or memory index, and vaccination can serve as effective control measures in combating this viral infection.

The nonlinear Schrödinger equation, held in high regard in the realms of plasma physics, fluid mechanics, and nonlinear optics, reverberates deeply within the field of ocean engineering, imparting profound insights across a plethora of phenomena. This article endeavours to establish a connection between the equation’s theoretical framework and its practical applications in ocean engineering, presenting a range of solutions tailored to grasp the intricacies of water wave propagation. By employing three methodologies, namely, the simplest equation method, the ratio technique, and the modified extended tanh-function method, we delineate various wave typologies, encompassing solitons and periodic manifestations. Enhanced by visual representations, our findings have the potential to deepen the comprehension of wave dynamics, with promising implications for the advancement of ocean engineering technologies and the refinement of marine architectural design.

In Sub-Saharan Africa, zoonotic diseases are the leading cause of sickness and mortality, yet preventing their spread has long been difficult. Vaccination initiatives have significantly reduced the frequency of zoonotic diseases mostly in African regions. Nonetheless, zoonotic illnesses continue to be a hazard to underdeveloped countries. Zoonotic infections are spread by direct contact, food, and water. We construct an epidemic model to understand zoonotic disease transmission phenomena. The model is examined using the fundamental results of fractional theory. The reproduction parameter ℛ0{{\mathcal{ {\mathcal R} }}}_{0} was obtained by inspecting the model’s steady states. The stability of the system’s steady states has been demonstrated. The system’s reproduction parameter is quantitatively explored by varying various input parameters. Furthermore, the presence and uniqueness of the solution of the proposed dynamics of zoonotic diseases have been demonstrated. Different simulations of the recommended zoonotic disease model with different input factors are performed to inspect the complex dynamics of zoonotic disease with the influence of various model factors. To establish effective prevention and control measures for the infection, we analyse dynamical behaviour of the system. Decreasing the fractional order θ\theta can decrease the infection level significantly. Different factors for reducing zoonotic diseases were recommended to regional policymakers.

This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u∗{u}^{\ast } is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type equation.

It is well known that viral infections have a high impact on public health in multiple ways, including disease burden, outbreaks and pandemic, economic consequences, emergency response, strain on healthcare systems, psychological and social effects, and the importance of vaccination. Mathematical models of viral infections help policymakers and researchers to understand how diseases can spread, predict the potential impact of interventions, and make informed decisions to control and manage outbreaks. In this work, we formulate a mathematical model for the transmission dynamics of COVID-19 in the framework of a fractional derivative. For the analysis of the recommended model, the fundamental concepts and results are presented. For the validity of the model, we have proven that the solutions of the recommended model are positive and bounded. The qualitative and quantitative analyses of the proposed dynamics have been carried out in this research work. To ensure the existence and uniqueness of the proposed COVID-19 dynamics, we employ fixed-point theorems such as Schaefer and Banach. In addition to this, we establish stability results for the system of COVID-19 infection through mathematical skills. To assess the influence of input parameters on the proposed dynamics of the infection, we analyzed the solution pathways using the Laplace Adomian decomposition approach. Moreover, we performed different simulations to conceptualize the role of input parameters on the dynamics of the infection. These simulations provide visualizations of key factors and aid public health officials in implementing effective measures to control the spread of the virus.

In this paper, we prove some common fixed point theorems on bicomplex metric space. Our results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.

A nonlinear viscoelastic wave equation with Balakrishnan-Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

In this work, we establish a theorem concerning the extension of positive weak solutions for a stationary fractional Laplacian problem featuring weight functions that change sign. Additionally, we introduce novel conditions to ensure the existence of a positive solution for the given problem. These conditions are derived utilizing the approach of sub-super solutions, thereby extending and complementing existing results in the literature. [ABSTRACT FROM AUTHOR]

Motivated by the idea which has been introduced by Boulaaras and Guefaifia [S. Boulaaras and R. Guefaifia, Existence of positive weak solutions for a class of Kirchhoff elliptic systems with multiple parameters, Math. Methods Appl. Sci. 41 2018, 13, 5203–5210] and by Afrouzi and Shakeri [G. A. Afrouzi, S. Shakeri and N. T. Chung, Existence of positive solutions for variable exponent elliptic systems with multiple parameters, Afr. Mat. 26 2015, 1–2, 159–168] combined with some properties of Kirchhoff-type operators, we prove the existence of positive solutions for a new class of nonlocal p(x)-Kirchhoff parabolic systems by using the sub- and super-solutions concept. [ABSTRACT FROM AUTHOR]

The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter ζ {{\zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following p ⁢ (x) {p(x)} -Kirchhoff system: { - M ⁢ (I 0 ⁢ (u)) ⁢ △ p ⁢ (x) ⁢ u = λ p ⁢ (x) ⁢ [ λ 1 ⁢ f ⁢ (v) + μ 1 ⁢ h ⁢ (u) ] in ⁢ Ω , - M ⁢ (I 0 ⁢ (v)) ⁢ △ p ⁢ (x) ⁢ v = λ p ⁢ (x) ⁢ [ λ 2 ⁢ g ⁢ (u) + μ 2 ⁢ τ ⁢ (v) ] in ⁢ Ω , u = v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with C 2 {C^{2}} boundary ∂ ⁡ Ω {\partial\Omega} , △ p ⁢ (x) ⁢ u = div ⁡ ( | ∇ ⁡ u | p ⁢ (x) - 2 ⁢ ∇ ⁡ u) {\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)} , p ⁢ (x) ∈ C 1 ⁢ (Ω ¯) {p(x)\in C^{1}(\overline{\Omega})} , with 1 < p ⁢ (x) {1