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In this paper, an accurate and reliable fully discretized linear numerical scheme has been developed for solving two-dimensional nonlinear coupled Burgers equation given by pre-defined initial and boundary values. The scheme is constructed based on the weighted finite differences utilizing a Richtmyer-type linearization in place of nonlinear terms appearing in the two-dimensional nonlinear coupled Burgers equation. It should be known that this linearization is used for the first time in this study for a two-dimensional nonlinear coupled partial differential equation. The performance and reliability of the present method are verified using four different widely-used examples, three of which have exact solutions and one has no an exact solution. To make sure that the current scheme exhibits good results, some error norms and convergence rate of the obtained approximate numerical solutions for each example are calculated and a comparison is made with other ones existing in the literature for the same parameter values. It has also been shown that the scheme is unconditionally stable for $0.5\le \theta \le 1$.

In this research, the integrable Zhanbota-IIA equation which plays an important role in nonlinear optical dynamics is analytically investigated. Based on the extended sinh-Gordon equation expansion method, its bright, dark, combined bright-dark, singular soliton, combined singular soliton and singular periodic solutions are constructed under some constraint conditions. In addition, by using the modified Jacobi elliptic function expansion method, the Jacobi elliptic function solutions and the other soliton solutions are derived. Further, the modulational instability is studied based on the standard linear stability analysis. The graphical interpretations of the obtained results are demonstrated graphically. To the best of our knowledge, this study represents a unique and significant contribution to investigating the integrable Zhanbota-IIA equation. In addition, the extracted soliton solutions open doors to understanding the nonlinear systems more deeply and connecting their potential across a range of scientific and technological domains.

This study investigates a cubic-quartic nonlinear Schrödinger model with third- and fourth-order dispersions without the disturbance in parabolic law media and group velocity dispersion (GVD). The analytical method is used to produce traveling wave solutions because the inverse scattering transform is unable to solve the Cauchy problem for this equation. The utilized analytical approach is novel and providing generalized families of solutions as compared to other, thus, prior to this study, there was not existing any work in which suck kinds of solutions were exist. The propagating solitons are driven using the unified auxiliary equation approach. As a result, the aforementioned technique is learned in the following areas: singular soliton, shock solution, mixed-singular, mixed trigonometric, mixed shock single solution, plane solutions, exponential, trigonometry, mixed-hyperbolic solution, periodic and mixed periodic solutions, and trigonometry. The stability of the topic under consideration is shown by the modulational instability (MI) study. The appropriate parametric values are shown alongside the graphical representation.

This work analyzes the ($3+1$)-dimensional Hirota-bilinear-like model (HBLM) analytically using the unified Riccati equation expansion approach (UREEA) and image encryption–decryption technique. We obtain various solutions for solitons, including dark, singular, periodic, and planar waves. We provide the contour, three-dimensional, and two-dimensional graphs of specific solutions that were successfully generated in the precise range space. The novelty of our work is to obtain the HBLM wave solutions that have never been discussed in the literature. Furthermore, image encryption and decryption techniques secure the wave transmission of ultra-short waves in an ocean. In addition, the stability analysis of the solutions is performed to ensure that the solutions are stable. Our findings are distinctive and fascinating and provide valuable mathematical insights for understanding the proposed model.

During worldwide epidemic of COVID-1, people started to depend on social media apps to cure their boredom. It was beneficial to provide information easily to a wide audience. The increased time people spent on their phones led to many, including teenagers and children, becoming addicted to social media, particularly TikTok. As a result, this usage of TikTok became an epidemic itself. Therefore, in this work, a fractional mathematical model is being developed to analyze the influence of TikTok on human population of different ages. In this paper, two equilibrium points for this model will be discussed and the theoretical stability will be proven. For numerical analysis, fractional Euler’s method will be used and the results will be discussed graphically and which group of population will be affected the most by frequent use of TikTok.

This paper extracts entirely new complex logarithmic prototypes to the nonlinear Black–Scholes equation used to investigate the price in economy. Newly submitted to the literature, $(\frac{1}{G^{\prime }})$-expansion method is used, successfully. Complex, logarithmic and hyperbolic non-traveling wave solutions are extracted to the considered model. Examined is the solution’s response for various conformable operator values. To verify the sensitivity of the numerical outcomes, tables with the stability and absolute error analysis findings are also provided. Von Neumann Stability analysis looks at the circumstances in which the model’s numerical findings are stable. Critical features, such as stability analysis and error evaluation, are presented, providing comprehensive information on the reliability and accuracy of the method. Errors that arise during the approximation process and the impact of parameter values are displayed through tables and graphs when employing the numerical technique. It offers a thorough comprehension of the financial phenomenon that the model under examination represents.

The aim of this work is to develop a model of alcohol that takes drug features into account. We assess the model feasibility and explain its formulation in terms of a nonlinear differential equation. Using the subsequent matrix generation technique, we ascertain the reproductive number in order to assess the dynamics of the model. We also examine the system equilibrium points, namely the positive and free alcohol equilibrium points. To gain insights into the stability properties of the model, we utilize the Lyapunov function and the Routh–Hurwitz criterion. Through these methods, we investigate both the local stability and global stability of the considered model. Furthermore, we employ numerical simulations to complement and illustrate the theoretical results obtained. These simulations provide visual representations that enhance the understanding of the model dynamics and behavior.

This work investigates the evolution of nonlinear eigenmodes in one-dimensional (1D) and two-dimensional (2D) optical systems with $(\mathcal{𝒫}\mathcal{𝒯})$-symmetric sinusoidal complex potentials, which supports real eigenvalue spectra up to the symmetry-breaking threshold. Numerical results show that in 1D and 2D systems, when nonlinearity increases, the threshold potential decreases. However, a deeper real potential increases the threshold in 1D, while it destabilizes the 2D optical beam as illustrated in the stability charts. Below this threshold, eigenfunctions exhibit symmetric behavior with smooth phase variations and no abrupt changes in energy flux. In contrast, the broken $(\mathcal{𝒫}\mathcal{𝒯})$-symmetric phase results in asymmetric eigenfunctions, indicating the instability. Stability analysis using Bogoliubov–de Gennes (BDG) matrix method and the propagation dynamics confirmed that eigenstates remain stable under small perturbations below the threshold but lose stability once the imaginary potential exceeds the threshold. Understanding the behavior of eigenmodes in $(\mathcal{𝒫}\mathcal{𝒯})$-symmetric potentials enables advanced control over beam profiles, potentially benefiting high-precision sensors, lasers, and optical trapping.

This paper proposed a single-degree-of-freedom self-sustained nonlinear oscillator capable of precisely predicting the bouncing force induced by a person during bouncing activity on a flat and rigid surface. A bouncing person produces essential internal energy required to maintain its motion, so it can be modeled as a self-sustained oscillator that can generate (i) the stable limit cycle, (ii) the periodic bouncing force signal, and (iii) the self-sustained motion. A hybrid Van der Pol–Rayleigh oscillator added with two quadratic and one cubic nonlinear terms has been derived to yield desired softening and hardening effects as well as even and odd harmonics, as observed from the analysis of experimental bouncing force data. The force applied on the surface corresponds to its restoring force. The stability analysis of the oscillator has been performed using the energy balance and perturbation methods. The model parameters are extracted from the experimental bouncing force data resulting from a bouncing test on a group of seven subjects with shoe insoles at six different frequencies guided by a metronome. The bootstrapping method has been performed to examine the convergence of mean values of each model parameter by increasing the cardinality of the experimental set. The bouncing force signals produced by the proposed model and experimental results demonstrate an excellent agreement.

The spinning circular solar sail is a promising spacecraft for long-duration missions. This work reveals its structural dynamic and stability behavior under the periodically time-varying solar radiation pressure and gravitational force. The geometric stiffness generated by the centrifugal force due to spinning and the coupling effect between the deformation and solar radiation pressure are taken into account. The von Kármán plate theory is adopted by neglecting the high-frequency in-plane vibrations and considering the effect of the in-plane internal force on the transverse vibration. The partial differential equation of the spinning solar sail is derived and further spatially discretized into periodically time-varying equations of motion. Effects of Poisson ratio and radius ratio on natural frequencies and mode shapes are analyzed, and curve veering phenomena are then observed. Steady-state periodic responses of the solar sail under the solar radiation pressure with different orbit distances, incident angles, and spinning angular velocities are analyzed. The stability analysis is rigorously performed by the Floquet theory rather than the commonly used approach of conducting the eigenvalue analysis at a series of specific discrete time nodes. Moreover, the stability boundary associated with transverse vibrations is determined, which contributes to the parameter design of the spinning solar sail.

The Ram Air Turbine (RAT), an emergency device installed within the aircraft, is designed to supply essential power after the failure of all other power systems. However, the design and development of RAT face significant dynamic challenges, particularly the instability of turbine speed in high wind speed environments. Assuming constant blade design parameters and fixed operating conditions with unchanged aerodynamic loads, a stability study of the centrifugal speed governing mechanism within the RAT is conducted. First, a dynamic model of the RAT is established, and numerical simulations are carried out to examine the variations in turbine speed and blade pitch angle. Subsequently, the process of the speed governing mechanism transitioning from an equilibrium state to being disturbed by load fluctuations is linearly approximated, enabling its control system to be modeled, and the aerodynamic load characteristics based on the speed regulation principle are integrated. Additionally, a stability criterion for speed is proposed using the Routh–Hurwitz criterion. Various factors affecting the stability of the speed regulation system are explored based on the dynamic and control models. By analyzing the stability domain of the RAT under different parameters, the influence of speed regulation system parameters on speed stability is assessed, revealing underlying principles and patterns. The results demonstrate that optimizing the design parameters of damping, springs, and centrifugal components in the mechanism significantly enhances the stability of the RAT speed governing mechanism, improving its adaptability to external aerodynamic load characteristics.

This paper presents a new, efficient, accurate, and unconditionally stable second-order time-stepping method for the incompressible thermal micropolar Navier–Stokes equations (TMNSE) using mixed finite elements. The method linearizes the nonlinear convective terms in the momentum equation, microrotation equation, and temperature equation, requiring the solution of a linear problem at each time step. The discrete curvature of the solution is added as a stabilizing term for linear velocity $u$, microrotation velocity $w$, pressure $p$, and temperature $T$ in the equations, respectively. Curvature stabilization ($u^{n+1}-2u^n+u^{n-1}$) is a new concept in computational fluid dynamics (CFD) aimed at improving the commonly used velocity stabilization ($u^{n+1}-u^n$), which only has first-order time accuracy and has adverse effects on important flow quantities such as drag coefficients. We derive a priori error estimates for the fully discrete linear extrapolation curvature stabilization method. The theoretical results and effectiveness of the new method are verified through a series of numerical experiments for $𝜃=\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, and $1$ in 2D and 3D, respectively. In particular, this work considers the thermal cavity-driven flow experiment to validate the numerical scheme and obtains good results.

We develop two different singularity-free interior stellar models characterizing anisotropic fluid distribution in this paper in the background of $f(R,T)$ gravity. The modified Einstein field equations and the corresponding pressure anisotropy are then calculated in conjunction with a static spherical spacetime. We then address the field equations by using two different constraints that make a system easy to solve. By taking into account specific forms of pressure anisotropy, we formulate two different stellar models. The differential equations appear in both cases whose solutions incorporate integration constants and we determine them by equating the metric potentials of the interior and the Schwarzschild exterior metrics at the spherical interface. Another condition that plays a crucial role in this regard is the vanishing radial pressure at the matching surface. We subsequently discuss multiple conditions that, when met, yield physically feasible compact models. We also consider the estimated data of a pulsar LMC X-4 along with five distinct values of the model parameter to graphically assess the developed solutions. It is concluded that both our models are well-aligned with the physically existence conditions in this modified gravity framework.

This paper explores a comprehensive approach to modeling compact stars that incorporates both normal matter and dark energy. We employ the Durgapal–Fuloria ansatz within the context of Rastall gravity to derive a relativistic analytical solution. The model is thoroughly analyzed both analytically and graphically, to assess its physical properties and facilitates a comparison with the results of classical general relativity. Our findings demonstrate that the model we have put forward is in substantial agreement with observational data for three different compact star representatives like Her X-1, PSR J0348+0432, and RX J1856.3-37.2. We evaluate the model’s viability by examining its energy conditions, stability, and adherence to the Buchdahl limit, all of which are found to be satisfactory. The analysis confirms the stable solution for Rastall parameter spanning −0.005 to 0.11, and it converges to standard general relativity when the coupling parameter approaches to zero.

This study introduces a novel fractional-order model to investigate the interplay between cancer and obesity and their treatment. Initially, we examine the solution’s existence and uniqueness for the proposed model. Additionally, we establish the boundedness of these solutions. Subsequently, we identify some potential equilibrium points of the cancer–obesity model and investigate their stability. To address the considered model, we propose fractional Euler’s and Adam’s methods. Theoretical and numerical analyses are conducted to assess the error estimates and performance of both methods with varying fractional-order derivatives. Moreover, we formulate an optimal control problem concerning cancer density and drug concentration. We delve into the existence of control and explore the first-order optimality conditions. We validate the analytical findings through numerical computations, demonstrating that administering drugs with control variables enhance immunity levels and reduce the burden of cancer.

Recent SARS-CoV-2 XBB and BQ subvariants exhibit noticeably enhanced resistance to neutralizing antibodies, including in those persons who are vaccinated and have had breakthrough Omicron infection or who have received the bivalent mRNA booster. Nonlinear dynamics is an intriguing technique for explaining the dynamical behaviors of COVID-19 illness. The goal of this study is to create a novel compartment mathematical model that takes into account populations that have received immunizations against new COVID-19 variants, particularly Omicron. The foundational characteristics of the model, including boundedness and positivity of solutions are established. Following that, the reproduction number has been used to illustrate the equilibrium analysis of steady states. The numerical design of the model has been introduced to assess how various control regimes affect COVID-19 dynamics. All control techniques have been found to have a positive effect on COVID-19 reduction.

In the entire world, pneumonia is one of the leading causes of death, which is particularly dangerous for young children (those under five years old) and the elderly (those over 65). A deterministic susceptible, vaccinated, exposed, infected, and recovered (SVEIR) model is used in this work to mathematically study the dynamics of pneumonia disease and examine stability analysis, basic reproduction numbers, and equilibrium points of dynamical systems theory models. Spatial equilibria are studied to model disease-free equilibria that are locally asymptotic stable. Numerical simulations of the model have been carried out using MATLAB21. The SVEIR flow and its variables for different parameter sets have been studied through numerical simulations. The solution to the issue is provided through the use of illustrated and explicated results. According to research findings, if vaccination rates rise over the necessary vaccination ratio, the sickness will finally vanish from the community.

Improving epidemic models to better reflect reality has long been a prominent concern for governments and researchers. This paper presents a novel Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model for human populations, offering a comprehensive analysis. The proposed model introduces a generalized SIRS epidemics framework encompassing three propagation scenarios. The paper establishes the positivity and boundedness of the system and demonstrates the stability of its equilibrium points. Furthermore, a controlled system is introduced, accompanied by three suggested control strategies to minimize the infected population while optimizing cost. To validate the analytical findings, a numerical example is provided. The paper concludes with a summary and outlines future research directions.

In this work, we formulate a two-strain model of dengue infection, incorporating the possibility of co-infection through integer and Caputo-type fractional derivatives. We present the foundational theory and results related to fractional operators for the analysis of the proposed dengue model. We establish the positivity and boundedness of the solution to validate the model. It is demonstrated that the solution to the proposed model exists and is unique. The basic reproduction number, denoted by ${{ℛ}}_c$, is determined, and the local asymptotic stability of the dengue-free equilibrium point is established when ${{ℛ}}_c<1$. We also prove the existence of an endemic equilibrium and establish the global stability of solutions to the fractional model in the Ulam–Hyers sense. Additionally, we construct a numerical scheme and prove its stability under certain conditions. Several numerical simulations are performed to validate our analytical results, and we examine the impact of various input factors on the solution pathways of the system through these simulations.

One of the most important factors influencing animal growth is non-genetics, which includes factors like nutrition, management and environmental conditions. By consuming their prey, predators can directly affect their ecology and evolution, but they can also have an indirect impact by affecting their prey’s nutrition and reproduction. Preys used to change their habitats, their foraging and vigilance habits as anti-predator responses. Cooperation during hunting by the predators develops significant fear in their prey which indirectly affects their nutrition. In this work, we propose a two-species stage-structured predator–prey system where the prey are classified into juvenile and mature prey. We assume that the conversion of juvenile prey to matured prey is affected by the fear of predation risk. Non-negativity and boundedness of the solutions are demonstrated theoretically. All the biologically feasible equilibrium states are determined, and their stabilities are analyzed. The role of various important factors, e.g. hunting cooperation rate, predation rate and rate of fear, on the system dynamics is discussed. To visualize the dynamical behavior of the system, extensive numerical experiments are performed by using MATLAB and MatCont 7.3. Finally, the proposed model is extended into a harvesting model under quadratic harvesting strategy and the associated control problem has been analyzed for optimal harvesting.