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Securing computers is crucial to prevent data breaches, identity theft, and financial losses. Virus incursions disrupt operations, causing downtime and costly repairs. Protective measures, including anti-virus software and cybersecurity practices, maintain network integrity and reduce the spread of malware. Combining robust cybersecurity with green computing strategies ensures efficient energy usage and sustainable network environments, safeguarding against viruses while contributing to both security and environmental goals In this study, we explore the dynamic behaviors of a modified version of the computer virus model and elucidate the connection between its parameters and network attributes. We employ Banach’s and Schaefer’s fixed-point theorems to assess the existence and uniqueness of solutions of the suggested model. Furthermore, we establish sufficient conditions for Ulam–Hyers stability within the envisioned computer virus model. To analyze solution trajectories and the impact of various input factors on computer virus dynamics, we utilize an efficient numerical technique, providing insight into the relationships between model parameters and enabling the design of networks that minimize the risk of virus outbreaks under various bifurcation conditions.

Viral infections pose significant threats to public health globally. Understanding the behavior, transmission, and epidemiology of viruses is essential for developing strategies to prevent, control, and manage outbreaks. Mathematical models help in identifying emerging viral pathogens, assessing their risks, and implementing effective public health measures to mitigate their impact. In this work, we formulate the dynamics of Covid-19 viral infection with the effect of vaccination in fractional framework. Our study is mainly concerned with the dynamical behavior and qualitative analysis of Covid-19 dynamics. The model is investigated for basic properties and the threshold of the system is determined. To scrutinize the solution of the recommend system, we use the fixed point theorems of Schaefer and Banach to evaluate the existence and uniqueness of solutions. Moreover, sufficient conditions for the Ulam–Hyers stability of system is established through mathematical skills. We examine the solution pathways of our model through a numerical scheme to show the importance different input parameters of the system. Our findings emphasize the pivotal role of asymptomatic carriers and losing rate of immunity as critical determinants that can heighten the challenge of controlling Covid-19. Vaccination rate and the fractional parameters are attractive parameters while asymptotic fraction poses a significant risk since it has the ability to spread the illness to uninfected areas. We suggest that effective control of the transmission rate can effectively regulate the intensity of Covid-19 transmission.

Abstract In this paper, we investigate a scenario concerning a coupled nonlocal singular viscoelastic equation with sources and distributed delay terms. By establishing suitable conditions, we have proved that a finite-time blow-up occurs in the solution.

The infection caused by Rift Valley fever (RVF) virus is a dangerous vector-borne disease found in humans, domestic, and wild animals. It is transferred through insect vectors to ruminant host and then spread through direct contact of infected animals with their body fluid or organs. In this paper, a fractal-fractional model for the transmission of RVF in the Caputo's sense was presented. We analyzed the model and determined the basic reproduction number through the next-generation matrix technique, indicated by $ \mathcal{R}_0 $. The global sensitivity technique is used for the sensitivity test of $ \mathcal{R}_0 $ to find out the most sensitive input-factors to the reproduction parameter $ \mathcal{R}_0 $. The existence and uniqueness results of the proposed fractal-fractional model were established. Then, we presented the fractal-fractional dynamics of the proposed RVF model through a novel numerical scheme under the fractal-fractional Caputo operator. In the end, the recommended model of RVF was highlighted numerically with the variation of different input parameters of the system. The key factors of the system were highlighted to the policymakers for the control and prevention of the infection.

This paper introduces a new study that examines the unique and analytical nature of the fractional solution to a fractional electromagnetic boundary value problem (BVP). This specific BVP is characterized by defining the tangential electromagnetic components. It has been proven that the analytical expressions for the fractional electromagnetic fields $ E^{\alpha} $, $ E^{*\alpha} $, $ H^{\alpha} $, and $ H^{*\alpha} $ do not vanish in any subregions $ \Omega_o^\alpha $ or $ \Omega^\alpha-\Omega_o^\alpha $. Furthermore, the unique solution makes $ E^{\alpha} = E^{*\alpha} $ and $ H^{\alpha} = H^{*\alpha} $ without singular fields at same region of the space. Analyticity of the fractional time-harmonic electromagnetic field within lossy or lossless dielectric regions is proven.

Abstract In this work, we consider a quasilinear system of viscoelastic equations with dispersion, source, distributed delay, and variable exponents. Under a suitable hypothesis the blow-up and growth of solutions are proved, and by using an integral inequality due to Komornik the general decay result is obtained in the case of absence of the source term f 1 = f 2 = 0 $f_{1}=f_{2}=0$ .

The mathematical modeling and dynamic analysis of time-delayed enzymatic chemical reactions in biological systems are presented in this research. The objective is to examine the function of time lags in these reactions and to get a complete knowledge of the behavior of biological systems in a reaction to modifications in the quantity present of reactants and products. The model, which is based on delay differential equations, includes a time delay term to account for the lag between changes in the concentration of reactants, reaction rate constants and product responses. The findings give insight into how enzymatic processes behave dynamically and how stability is impacted by time lags, oscillation and general efficiency of the system. These results have significant importance for our comprehension of how biological processes are regulated and for the creation of biological control structures.

Impulse control problems pose significant challenges in decision making under uncertainty, particularly in scenarios where actions must be taken at irregular intervals or in response to abrupt changes in the environment. Ensuring both the existence and uniqueness of optimal control policies in such settings is paramount for practical implementation and theoretical understanding. In this study, we investigate the existence and uniqueness of solutions for impulse control problems utilizing asynchronous algorithms. This study presents a novel demonstration concerning the uniqueness and existence of Partially Quasi-Variational Inequalities (PQVIs). Our approach unfolds through a four-step methodology, leveraging distinctive features of a discrete iterative technique. This methodology integrates semi-implicit techniques with respect to the variable t and employs spatial approximation via finite element methods. We establish a pivotal connection between a fixed-point mapping and the discrete EQVIs system, utilizing it to formulate a discrete algorithm featuring a semi-implicit time scheme. Furthermore, we introduce a monotone iterative scheme inspired by Bensoussan’s algorithm and rigorously confirm several of its properties through mathematical proofs. This work contributes to advancing the understanding and application of iterative techniques in solving PQVIs, offering insights into their uniqueness and existence.

In this study, we present two meshless schemes, namely the radial basis function (RBF) method and the polynomial method, for the numerical investigation of the time-fractional Harry Dym equation and the Drinfeld-Sokolov-Wilson system. In both methods, the temporal derivatives are estimated using the Caputo operator, while the spatial derivatives are approximated either through radial basis functions or polynomials. Additionally, a collocation approach is employed to convert the system of equations into a system of linear equations that is easier to solve. The accuracy of the methods is assessed by calculating the $ L_{\infty} $ error norm, and the outcomes are displayed through tables and figures. The simulation results indicate that both methods exhibit strong performance in handling the fractional partial differential equations (PDEs) under investigation.

It is acknowledged that dengue infection has a significant economic impact due to healthcare costs and lost productivity. Research can provide insights into the economic burden of the disease, guiding policymakers in their allocation of resources for prevention and control interventions. In this work, we structured a novel mathematical model that describes the spread of dengue with the effects of carriers, an index of memory and vaccination. To show the effect of treatment on the dynamics of dengue, we have incorporated medication-related treatment into the system. The proposed dynamics are represented by using fractional derivatives to capture the role of memory in the control of the infection. We introduced the fundamental principles and notions of non-integer derivatives for the analysis of the model; moreover, the existence and uniqueness results for the solution of the system have been established with the help of mathematical skills. The theory of fixed points has been utilized for the analysis and examination of the system. We have established Ulam-Hyers stability for the recommended system of dengue infection. Regarding the numerical findings, a numerical method is presented to highlight the solution pathways for the system of dengue infection. Several simulations have been performed to visualize the contribution of the input parameters of the system to the prevention and control of the infection. The index of memory, vaccination, and treatment are suggested to be attractive parameters which can reduce the level of infection while the biting rate, asymptomatic carriers and transmission rate are critical as they can increase the risk of the infection in society. Our findings not only provide information for the effective management of the infection they also possess valuable insights that can improve public health.

Abstract We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.

Abstract The infection of human papilloma virus (HPV) poses a global public health challenge, particularly in regions with limited access to health care and preventive measures, contributing to health disparities and increased disease burden. In this research work, we present a new model to explore the transmission dynamics of HPV infection, incorporating the impact of vaccination through the Atangana–Baleanu derivative. We establish the positivity and uniqueness of the solution for the proposed model HPV infection. The threshold parameter is determined through the next-generation matrix method, symbolized by $${\mathcal {R}}_0$$ R 0 . Moreover, we investigate the local asymptotic stability of the infection-free steady-state of the system. The existence of the solutions of the recommended model is determined through fixed-point theory. A numerical scheme is presented to visualize the dynamical behavior of the system with variation of input factors. We have shown the impact of input parameters on the dynamics of the system through numerical simulations. The findings of our investigation delineated the principal parameters exerting significant influence for the control and prevention of HPV infection.

This study examines the general decomposition and global existence of solutions of the nonlinear wave equation, considering the incorporation of fractional derivative boundary conditions and memory components.

In society, there are numerous hurdles hindering the advancement of women. Additionally, violence against women remains a significant issue, exacerbating their challenges. Therefore, addressing these violence-related challenges in a mature manner is crucial to creating a comfortable environment for women to thrive. In this paper, we present a nonlinear mathematical model focusing on women’s issues and empowerment as key factors. The model exhibits two equilibrium points: a violence-free equilibrium point and a violence-present equilibrium point. We analyze the stability of these equilibrium points and explore the influence of parameters within the model. Furthermore, we extend the model to incorporate stochastic effects. Subsequently, we compare the results of the stochastic model with those of the deterministic model. Our comparison reveals that the stochastic and deterministic models exhibit similar behaviors, albeit with slight oscillations in stochastic populations compared to deterministic ones.

In this research work, we introduced novel distance measure in picture fuzzy sets and prove all the triangle inequality associated with it. There is an introduction to the use of the distance measure in medical diagnostics. In addition to this, we have shown a comparison between two distance functions ρ and ρ∗ in this research work. Our observation indicates that the novel fuzzy distance measure ρ∗ exhibits greater efficacy in the context of disease diagnosis compared to the conventional measure ρ.

An analytical investigation has been conducted on the steady-state conduction and a diluted gas exposed to a thermal radiation field experiences radiative heat transfer. Analytical solutions have been derived for the transport Boltzmann BGK partial differential equations system. The investigation delves into the irreversible thermodynamic behavior of the system through the utilization of the Liu-Lees model, which employs two-stream Maxwellian distribution functions. The moment approach is utilized to predict the behavior of macroscopic gas parameters like temperature, concentration, and Fourier heat flux. This process entails substituting these parameters into the two-stream Maxwellian distribution functions to analyze the system's non-equilibrium thermodynamic characteristics, which encompass gas particles and the heated plate. Through this analysis, kinetic coefficients, entropy, entropy flux, entropy production, and thermodynamic forces are determined. The investigation assesses the validation of Onsager's reciprocity relation, the Boltzmann H-theorem, the Le Chatelier principle, and the second law of thermodynamics within the system. Furthermore, the Gibbs formula is employed to estimate the ratios between various contributions to internal energy changes, which are determined by the total derivatives of extensive parameters. The non-equilibrium velocity distribution function was calculated for the first time, and its behavior was compared with that of the equilibrium one using 3D graphics. The behavior of the computed variables is predicted through graphical analysis and discussion of the results.

The problem of existence and uniqueness of nonlinear Goursat is studied by Claude Wagschal in different spaces: holomorphic spaces, partially holomorphic, continuous-Gevrey and Gevrey-holomorphic. In this paper, we replace the sequence (n!)d−1 in the case Gevery with suitable sequence (Mn) in the case Carleman. After that, we study the problem of existence and uniqueness of nonlinear Goursat problem in Denjoy–Carleman space.

Abstract An enormous cost is placed on people, communities, and healthcare systems by bacterial infections. Measures of the burden of bacterial infections include morbidity, mortality, economic expenditures, and overall effects on public health. Campylobacteriosis is a bacterial infection imposes a significant economic burden on both individuals and societies due to its prevalence, healthcare costs, and the associated loss of productivity. In our research, we develop a model to analyze the transmission of campylobacteriosis infection, taking into account factors such as vaccination and treatment. We also examine the fundamental characteristics of fractional calculus to understand the model better. The equilibria of the model are studied, and we calculate the reproduction parameter denoted as $$\mathcal{R}_{0}$$ R 0 . Furthermore, we provide proof of stability for the equilibria of the system. Lastly, we conduct numerical investigations to explore the variation of the system’s reproduction parameter with different input parameters. We have established the necessary conditions to ensure the existence and uniqueness of solutions for the proposed model of campylobacteriosis infection. To better understand the complex dynamics of campylobacteriosis infection, we conduct various simulations of the suggested model while modifying the input factors. These simulations allow us to investigate the effects of different input parameters on the dynamics of campylobacteriosis infection. We analyze the dynamic behavior of the system to develop efficient control strategies for managing the infection. Notable improvements have been observed by reducing the order of the fractional derivative. Based on our findings, we propose various factors to the policy makers in the community to mitigate the spread of campylobacteriosis infection.

It is significant to investigate the transmission dynamics of vector-borne infection because it has a global impact, can help predict and prevent future outbreaks, is important for understanding the impact of climate change on public health, can lead to more effective control strategies, and can improve our comprehension of the dynamics of these infections. Our paper presents a new model for chikungunya virus infection, which considers treatment and vaccination, using the Atangana-Baleanu derivative within the framework of the Caputo definition. First of all, we examine the positivity and uniqueness of the solution for the model. Then, we find out the fundamental results of the suggested model, such as the steady-state without infection and the R0 value that indicates endemicity pertaining to the system. Furthermore, we verify the local asymptotic stability of the infection-free steady-state. Through the application of fixed-point theory, we demonstrate the existence of the solution and propose a numerical methodology to investigate the dynamic behavior of the model. Finally, we performed simulations to demonstrate the effects of vaccination, transmission probability, treatment, and index of memory on the system's solution pathways. The findings of this research suggest the most sensitive parameters of the system responsible for the control and prevention of the infection. Emphasizing the significance of the memory index as an attractive parameter, we propose its consideration to the policy makers as a control parameter for the prevention of the infection.

In this study, we formulate a mathematical model in the framework of the Atangana-Baleanu fractional derivative in Caputo sense to study the transmission of tungiasis. In this formulation, interactions between the human host and the sand fleas are taken into consideration, including factors like infestation rate, incubation duration, and recovery rate. We calculate the basic reproduction parameter for the system, symbolized by $\mathcal{R}_0$ with the help of the next-generation matrix technique. A novel numerical scheme for encapsulating the non-local and memory-dependent aspects of the system is conceptualized via the Atangana-Baleanu fractional derivative. We prove the existence and uniqueness of the solution of the model of the infection and establish stability of the steady-states of the model. In addition to this, numerical simulations are carried out to evaluate the efficiency of interventions like campaigns for better sanitation and treatment, and to investigate the influence of various management techniques on the prevalence of tungiasis. The outcomes of the numerical simulations give us information about the possible efficacy of different control strategies in lowering the incidence of tungiasis. This research gives quantitative tools to enhance decision-making processes in public health treatments and advances our understanding of the dynamics of the tungiasis.

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

In this study, we present a numerical method named the logarithmic non-polynomial spline method. This method combines conformable derivative, finite difference, and non-polynomial spline techniques to solve the nonlinear inhomogeneous time-fractional Burgers–Huxley equation. The developed numerical scheme is characterized by a sixth-order convergence and conditional stability. The accuracy of the method is demonstrated with 3D mesh plots, while the effects of time and fractional order are shown in 2D plots. Comparative evaluations with the cubic B-spline collocation method are provided. To illustrate the suitability and effectiveness of the proposed method, two examples are tested, with the results are evaluated using L2 and L∞ norms.

Substance abuse is considered to be predominantly a problem for all in society, and much substance abuse research is done with a male majority. Recent drug addiction studies, however, reveal that there are substantial gender variations in substance-related epidemiology. The epidemiology of women’s substance misuse raises issues distinct from those addressed by men’s substance abuse. Women with addictions are more likely than males to experience several impediments to accessing and entering substance misuse treatment. In this regard, we framed a mathematical model by considering the whole population as a women population. For this system, we found pair of equilibrium points namely the Addiction-free equilibrium point and the Non-trivial equilibrium point. Moreover, the addicts generation number (threshold parameter) RA has been found to investigate the spread of addiction. Also, the local and global stability analysis for both the equilibrium points have investigated. Further, to analyze the decrease and increase of the infected population and recovery population respectively, an optimum control analysis is done with two control parameters using Pontryagin’s maximum principle. The influence of addict generation numbers on significant parameters involved in RA has been shown. An optimum control study projected that social media awareness is substantially more efficient than the fixed control in optimizing drug addiction among women. Through numerical simulations we have exhibited, the effect of with and without control on each population. Finally, it produced positive efficacy by reducing the number of infected and increasing the rehabilitation population.

The study of liquid comprising nanoparticles for energy transmission characteristics has gained prominence due to its diverse real-life applications in different energy systems. Recently, scientists have been endeavoring to introduce a new type of liquid, despite the already proven effectiveness of Nano liquid's thermal efficiency. To meet the demands of various engineering and industrial applications, scientists have developed a new class of liquid called hybrid Nano liquid by dispersing two types of nano-sized particles in a regular liquid. Given the significance of hybrid Nano liquid in this study, we have considered (titanium dioxide, sodium alginate, silver) and (graphite oxide) in an MHD Casson fluid (bio-viscosity blood) near the lower stagnation point of a solid sphere. Additionally, this research addresses thermal radiations and the effects of magnetic fields. For the numerical results of the proposed ODEs, the Keller box method was employed. Engineering quantities obtained from this investigation are discussed in tables. From the observed graphs, it is evident that the velocity of the liquid near the lower stagnation point decreases as the magnetic effects increase.

The present work aims to investigate solitary wave solutions for two recently developed extended equations in the context of (2+1)-dimensional and (3+1)-dimensional structures. The equations under consideration are of the Korteweg–de Vries (KdV) type, which are well-recognized as significant aspects of fluid dynamics. These equations have broad applications in physics, mathematics, and other scientific disciplines, particularly in the study of waves, soliton theory, plasma physics, biology and chemistry, and nonlinear phenomena. Its soliton solutions and integrability properties make it a fundamental model in various areas of research. This serves as the main motivation for our research work. To analyze these equations, we employ an advanced direct algebraic equation method capable of generating several sorts of solutions, including solitary and shock wave solutions, as well as their combination. In addition to these wave phenomena, singular solitons and solutions expressed in Jacobi and Weierstrass doubly periodic types have also been observed. The utilization of this outstanding technique and the subsequent acquisition of novel solutions demonstrate the originality of our study. This also allows further exploration of nonlinear models that accurately depict significant physical processes in our everyday existence.

The primary objective of this research is to examine a nonlinear elastic membrane equation incorporating delay and source terms within a bounded domain. We obtain sufficient conditions on the initial data and the involved functionals for which the energy of solutions with non positive initial energy as well as positive initial energy blow up in a finite-time. In addition, this research work provides estimates for the lifespan of these solutions.

Abstract The burden of vector-borne infections is significant, particularly in low- and middle-income countries where vector populations are high and healthcare infrastructure may be inadequate. Further, studies are required to investigate the key factors of vector-borne infections to provide effective control measure. This study focuses on formulating a mathematical framework to characterize the spread of chikungunya infection in the presence of vaccines and treatments. The research is primarily dedicated to descriptive study and comprehension of dynamic behaviour of chikungunya dynamics. We use Banach’s and Schaefer’s fixed point theorems to investigate the existence and uniqueness of the suggested chikungunya framework resolution. Additionally, we confirm the Ulam–Hyers stability of the chikungunya system. To assess the impact of various parameters on the dynamics of chikungunya, we examine solution pathways using the Laplace-Adomian method of disintegration. Specifically, to visualise the impacts of fractional order, vaccination, bite rate and treatment computer algorithms are employed on the infection level of chikungunya. Our research identified the framework’s essential input settings for managing chikungunya infection. Notably, the intensity of chikungunya infection can be reduced by lowering mosquito bite rates in the affected area. On the other hand, vaccination, memory index or fractional order, and treatment could be used as efficient controlling variables.

Abstract The decay rate of solutions to a Bresse system in thermoelasticity of type III with respect to the distributed delay term is the subject of this study. We demonstrate our major finding utilising the energy approach in the Fourier space.

The utilization of time-fractional PDEs in diverse fields within science and technology has attracted significant interest from researchers. This paper presents a relatively new numerical approach aimed at solving two-term time-fractional PDE models in two and three dimensions. We combined the Liouville–Caputo fractional derivative scheme with the Strang splitting algorithm for the temporal component and employed a meshless technique for spatial derivatives utilizing Lucas and Fibonacci polynomials. The rising demand for meshless methods stems from their inherent mesh-free nature and suitability for higher dimensions. Moreover, this approach demonstrates the effective approximation of solutions across both regular and irregular domains. Error norms were used to assess the accuracy of the methodology across both regular and irregular domains. A comparative analysis was conducted between the exact solution and alternative numerical methods found in the contemporary literature. The findings demonstrate that our proposed approach exhibited better performance while demanding fewer computational resources.

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.

This study examines the pace at which solutions to a Bresse system in combination with the Cattaneo law of heat conduction and the dispersed delay term degradation. We establish our major finding utilizing the energy approach in the Fourier space.

In this work, we deal with a one-dimensional Cauchy problem in Timoshenko system with thermal effect and damping term. The heat conduction is given by the theory of Lord-Shulman. We prove that the dissipation induced by the coupling of the Timoshenko system with the heat conduction of Lord-Shulman's theory alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow.

The main goal of this research is to present the concept of enhancing heat transfer within emerging technology. To achieve this, tiny metal and nonmetal particles ranging from 1 to 100 nm in size are introduced into base liquids. These nanoscale particles are utilized to improve the thermal performance of the liquids, leading to what are termed nanofluids. The utilization of these fluids and the examination of the flow of thin films have valuable implications across various sectors such as engineering, technology, and industries. This research focuses on analyzing the convective flow behavior of nanofluids, specifically, graphene oxide-ethylene glycol (GO−EG) and graphene oxide-water (GO−W) on a moving surface. The study investigates the impacts of magnetic fields and varying viscosity. By making use of the thermophysical characteristics of the base fluid and the nanofluid, as well as implementing a similarity transformation within the fundamental equations that govern energy and momentum, we formulate a 5th order nonlinear ordinary differential equation (NODE) to describe the velocity profile. This is combined with a second-order NODE that describes the distribution of temperature. To solve this derived NODE, we employ a method known as the Homotopy Analysis Method (HAM) for analytical solution. The impact of the relevant factors, Prandtl number, including magnetic field parameter, thickness of the liquid, couple stress parameter, temperature distribution, dynamic viscosity, and Eckert number, on the skin friction, velocity profile, and Nusselt's number are interrogated through graphical representation. The velocity field exhibits a decline as the couple stress parameter, magnetic field parameter, liquid thickness, and dynamic viscosity experience an increase. Conversely, the temperature field displays a rise as the Eckert number and dynamic viscosity experience an increase. To ensure the convergence of the issue, dual solutions of the problem are employed, and this is verified through the utilization graphs and tables. Due to the considerable challenge encountered in heat transfer applications for cooling diverse equipment and devices across industries like automotive, microelectronics, defense, and manufacturing, there is a strong expectation that this theoretical methodology could make a favorable contribution towards enhancing heat transfer efficiency. This improvement is sought to meet the requirements of the manufacturing and engineering sectors.

In this work, we examine the existence and nonexistence of solution for lamé wave equation with a nonlinear source terms and fractional boundary dissipation. By using the semi group theory approach, we prove the existence of a global solution. Furthermore, we established the blow-up results under the non-positive initial energy condition.

This study examines Marangoni convection in blood-based carbon nanotubes nanofluid's stagnation point flow over a time-dependent stretching surface. This study is inspired by the emerging importance of nanofluids in a variety of scientific and technical fields due to their unique and varied uses and effective thermal activities. Some examples of potential applications of these fluids include cancer treatment, magnetic refrigeration, drug delivery, and magnetic resonance imaging. Two types of nanoparticles are considered name is single wall carbon nanotube and multi wall carbon nanotube blood is takin as base fluid. Nonlinear partial differential equations are used to simulate the specified flow issue using momentum and energy conservation principles. Using a similarity transformation, the resultant is transformed into nonlinear with reduced dimensions. The relations for velocity profile and temperature distribution are calculated from the developed nonlinear ordinary differential equation by using an approximate analytical technique called the homotopy asymptotic method. Subsequently, these equations are implemented and executed within Mathematica software. The investigation focuses on significant outcomes such as momentum filed, energy filed, Skin friction coefficients, and Nusselt number. Graphs are used to interpret the effects of several factors, including the Marangoni parameter, nanoparticle volume friction, stretching parameter, slip parameter, and Prandtl number. The behavior of Nusselt's number and skin friction coefficient was also checked with the help of graphs and tables. The convergence of the solutions is checked with the help of auxiliary functions as well.

In this research work, we construct an epidemic model to understand COVID-19 transmission vaccination and therapy considerations. The model's equilibria were examined, and the reproduction parameter was calculated via a next-generation matrix method, symbolized by $ \mathcal{R}_0 $. We have shown that the infection-free steady state of our system is locally asymptotically stable for $ \mathcal{R}_0 < 1 $. Also, the local asymptotic stability of the endemic steady state has been established for $ \mathcal{R}_0 > 1 $. We have used a partial rank correlation coefficient method for sensitivity analysis of the threshold parameter $ \mathcal{R}_0 $. The contribution of vaccination to the threshold parameter is explored through graphical results. In addition to this, the uniqueness and existence of the solution to the postulated model of COVID-19 infection is shown. We ran various simulations of the proposed COVID-19 dynamics with varied input parameters to scrutinize the complex dynamics of COVID-19 infection. We illustrated the variation in the dynamical behavior of the system with different values of the input parameters. The key factors of the system are visualized for the public health officials for the control of the infection.

It is valuable to measure the epidemiological significance of malaria, since there has been a growing interest in reducing malaria through improved local and national health care systems. We formulate the dynamics of malaria infection via a fractional framework to understand the intricate transmission route of malaria and to identify the role of memory for the control of malaria. The model is investigated for basic results, moreover, the basic reproduction number is determined symbolized by $ \mathcal{R}_0 $. We have shown the local stability of the disease-free steady-state of the system for for $ \mathcal{R}_0 < 1 $. The existence and uniqueness of the solution of the system are examined. The Adams Bashforth approach in fractional form is applied to analyse the numerical outcomes of the mathematical model. Furthermore, in order to realise more efficiently, the Atangana-Baleanu (ABC) fractional nonlocal operator, which was just invented, is used. The stability of the system is investigated through the fixed-point theorems of Krasnoselskii and Banach. The behaviour of the approximation solution is illustrated in terms of graphs across various fractional values and other factors of the systems. After all, a brief analysis of the simulation's findings is provided to explain how infection transmission dynamics occur in society.

One of the most harmful and widespread sexually transmitted diseases is syphilis. This infection is caused by the Treponema Palladum bacterium that spreads through sexual intercourse and is projected to affect $ 12 $ million people annually worldwide. In order to thoroughly examine the complex and all-encompassing dynamics of syphilis infection. In this article, we constructed the dynamics of syphilis using the fractional derivative of the Atangana-Baleanu for more accurate outcomes. The basic theory of non-integer derivative is illustrated for the examination of the recommended model. We determined the steady-states of the system and calculated the $ \mathcal{R}_{0} $ for the intended fractional model with the help of the next-generation method. The infection-free steady-state of the system is locally stable if $ \mathcal{R}_{0} < 1 $ through jacobian matrix method. The existence and uniqueness of the fractional order system are investigate by applying the fixed-point theory. The iterative solution of our model with fractional order was then carried out by utilising a newly generated numerical approach. Finally, numerical results are computed for various values of the factor $ \Phi $ and other parameters of the system. The solution pathways and chaotic phenomena of the system are highlighted. Our findings show that fractional order derivatives provide more precise and realistic information regarding the dynamics of syphilis infection.

Social media addiction (SMA) is the excessive use of social media platforms, resulting in negative consequences for individuals. It is characterized by an uncontrollable urge to use social media, leading to negative effects in human's life. This study aims to construct a mathematical model to conceptualize the transmission dynamics of SMA and explore the underlying mechanisms of this harmful addiction in the framework of fractional derivative. The fundamentals of fractional calculus are listed for examining the model. Equilibrium points are identified, and the reproduction parameter R0 is computed to understand the dynamics of SMA spread. Stability analysis of the equilibria is performed, and the impact of various input parameters is numerically investigated. The existence and uniqueness of the proposed SMA model are demonstrated through simulations, which also study the intricate dynamics with respect to different input factors. To develop effective control strategies, the system's dynamical behavior is examined, and the influence of fractional derivative order on fluctuations is explored. This research offers a range of suggestions aimed at reducing the occurrence of social media addiction.

An analysis of steady two-dimensional boundary layer MHD (magnetohydrodynamic) nanofluid flow with nonlinear thermal radiation across a horizontally moving thin needle was performed in this study. The flow along a thin needle is considered to be laminar and viscous. The Rosseland estimate is utilized to portray the radiation heat transition under the energy condition. Titanium dioxide (TiO$ _2 $) is applied as the nanofluid and water as the base fluid. The objective of this work was to study the effects of a magnetic field, thermal radiation, variable viscosity and thermal conductivity on MHD flow toward a porous thin needle. By using a suitable similarity transformation, the nonlinear governing PDEs are turned into a set of nonlinear ODEs which are then successfully solved by means of the homotopy analysis method using Mathematica software. The comparison result for some limited cases was achieved with earlier published data. The governing parameters were fixed values throughout the study, i.e., $ k_1 $ = 0.3, $ M $ = 0.6, $ F_r $ = 0.1, $ \delta_\mu $ = 0.3, $ \chi $ = 0.001, $ Pr $ = 0.7, $ Ec $ = 0.5, $ \theta_r $ = 0.1, $ \epsilon $ = 0.2, $ Rd $ = 0.4 and $ \delta_k $ = 0.1. After detailed analysis of the present work, it was discovered that the nanofluid flow diminishes with growth in the porosity parameter, variable viscosity parameter and magnetic parameter, while it upsurges when the rate of inertia increases. The thermal property enhances with the thermal conductivity parameter, radiation parameter, temperature ratio parameter and Eckert number, while it reduces with the Prandtl number and size of the needle. Moreover, skin friction of the nanofluid increases with corresponding growth in the magnetic parameter, porosity parameter and inertial parameter, while it reduces with growth in the velocity ratio parameter. The Nusselt number increases with increases in the values of the inertia parameter and Eckert number, while it decliens against a higher estimation of the Prandtl number and magnetic parameter. This study has a multiplicity of applications like petroleum products, nuclear waste disposal, magnetic cell separation, extrusion of a plastic sheet, cross-breed powered machines, grain storage, materials production, polymeric sheet, energy generation, drilling processes, continuous casting, submarines, wire coating, building design, geothermal power generations, lubrication, space equipment, biomedicine and cancer treatment.

The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $ g_{i} :{\bf R}_{+}\rightarrow {\bf R}_{+} $ holds true the below $ g_{i}'(t)\leq-\zeta_{i}(t)G_{i}(g_{i}(t)), \quad \forall t\in {\bf R}_{+}, \quad {\rm{for}} \quad i = 1, 2, $ in which $ \zeta_{i} $ and $ G_{i} $ are functions. We demonstrate the stability of the system under this highly generic assumptions on the behaviour of $ g_i $ at infinity and by dropping the boundedness assumptions in the historical data.

The Lord Shulman swelling porous thermo-elastic soil system with the effects of microtemperature, temperatures and distributed delay terms is considered in this study. The well-posedness result is established by the Lumer–Phillips corollary applied to the Hille–Yosida theorem. The exponential stability result is proven by the energy method under suitable assumptions.

In this work, by the use of a semigroup theory approach, we provide a global solution for an initial boundary value problem of the wave equation with logarithmic nonlinear source terms and fractional boundary dissipation. In addition to this, we establish a blow-up result for the solution under the condition of non-positive initial energy.

The heat transfer ratio plays an important role in the industrial and engineering sectors; in this model, the authors used the hybrid nanofluid because the heat transfer ratio of the hybrid nanofluid is more than that of the base fluid. The key objective of this research work is to boost up the heat transfer ratio, for example, not only the accomplishment of energy is enough but is also expected to regulate the feeding of energy, and this is possible only to approve the development of heat transmission liquids to the mechanism of the expenditures of energy and improvement. The current research study investigates the influence of Marangoni convection, solar radiation, and viscous dissipation on the bioconvection couple stress flow of the hybrid nanofluid over a shrinking surface. This type of flow has some important application in the industrial and engineering sectors for the purpose of cooling and heating effect. To transform the non-dimensionless form of the differential equation to the dimensionless form, the authors used the defined similarity transformation. The transformed dimensionless form of the differential equation is solved by the homotopic analysis method. The obtained important result is determined with the help of graphs which is obtained from velocity and temperature equations. The impression of different parameters such as couple stress parameter, Marangoni convection parameter, nanoparticle volume fraction, solar radiation parameter, magnetic field parameter, thermophoresis parameter, Eckert number, and Prandtl number is taken over graphs. The skin friction coefficient and Nusselt number are described in the form of tables.

Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by R0. The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if R0<1 and unstable otherwise. Furthermore, the Ulam–Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection.

The economic burden of HIV extends beyond the individual level and affects communities and countries. HIV can lead to decreased economic growth due to lost productivity and increased healthcare costs. In some countries, the HIV epidemic has led to a reduction in life expectancy, which can impact the overall quality of life and economic prosperity. Therefore, it is significant to investigate the intricate dynamics of this viral infection to know how the virus interacts with the immune system. In the current research, we will formulate the dynamics of HIV infection in the host body to conceptualize the interaction of T-cells and the immune system. The recommended model of HIV infection is presented with the help of fractional calculus for more precious outcomes. We introduce numerical methods to demonstrate how the input parameters affect the output of the system. The dynamical behavior and chaotic nature of the system are visualized with the variation of different input factors. The system’s tracking path has been numerically depicted and the impact of the viruses on T-cells has been demonstrated. In addition to this, the key factors of the system has been predicted through numerical findings. Our results predict that the strong non-linearity of the system is responsible for the chaos and oscillation, which are so closely related. The chaotic parameters of the system are highlighted and are recommended for the control of the chaos of the system.

In this work, we develop a sophisticated mathematical model for dengue transmission dynamics based on vaccination, reinfection and carrier class assumptions. We utilize Caputo-Fabrizio fractional framework to represent the transmission phenomena of dengue. For the model analysis, the basic theory and findings of the Caputo-Fabrizio fractional operator are provided. We utilize the next-generation approach to calculate the threshold parameter R0 for our proposed dengue infection model. We have shown that the disease-free steady-state of the hypothesized dengue system is locally asymptotically stable if R0

It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the system. Here, we use fractional derivative to structure tumor-immune interactions. Furthermore, in our research, we concentrated on the qualitative investigation and time series analysis of tumor-immune cell interactions. The solution routes are examined using a new numerical technique to emphasis the impact of the factors on tumor-immune system. We focused on the behaviour of the system with fluctuation of different values. The most crucial components of the proposed system are identified and policymakers are advised. The outcomes of the present study are the strong predictor of clinical success and the in-out of immune cells in a tumour is also critical to treatment efficacy. As a result, studying the behaviour of tumor-immune cell interactions is important to predict crucial factors for the prevention and management to the health officials.