Journal of Applied Mathematics

In this manuscript, we establish existence, uniqueness, and trajectory controllability for higher order noninstantaneous impulsive fractional neutral stochastic differential equations. First, solvability and uniqueness results are obtained using a fixed‐point approach with appropriate assumptions on nonlinear functions. Next, we deal with the strongest notion of controllability called the trajectory controllability for noninstantaneous impulsive fractional neutral stochastic differential equations via Gronwall’s inequality. The higher order fractional system with the deviating argument is the novel approach of this manuscript. Finally, a demonstration of an example yields theoretical outcomes.

Leptospirosis and melioidosis are emerging tropical diseases that are seriously affecting both human and animal populations worldwide. The actual incidence and fatal cases of the diseases are underreported due to a lack of awareness of the diseases, underuse of clinical microbiology laboratories test, and limitations of the model. In this paper, a new deterministic mathematical model for the coinfection of leptospirosis and melioidosis with optimal controls is presented. Based on the next‐generation matrix approach, the basic reproduction numbers for the coinfection model as well as for submodels are computed to analyze their dynamics behavior. The disease‐free equilibrium point of the melioidosis‐only submodel is proven to be globally asymptotically stable when the basic reproduction number ( R 0 m ) is less than unity, whereas the existence of its unique positive endemic equilibrium is shown if R 0 m > 1. Based on the center manifold theory, the endemic equilibrium point of the leptospirosis‐only submodel is proven to be locally asymptotically stable when the basic reproduction number ( R 0 l ) is greater than unity. The disease‐free equilibrium point of the full model is locally asymptotically stable whenever the basic reproduction number ( R 0 m l ) less than unity. Sensitivity analysis for the basic reproduction number of the model is performed to determine the most influencing parameters on the transmission dynamics of the model. Furthermore, the model was extended into an optimal control problem by incorporating four time‐dependent control functions. Pontryagin’s maximum principle was used to derive the optimality system for the optimal control problem. The optimality system was simulated using the forward–backward sweep method to show the effectiveness and cost‐effectiveness of different optimal control strategies in combating the burden of leptospirosis–melioidosis coinfection. The incremental cost‐effectiveness ratio was applied to determine the most cost‐effective strategy. The numerical results revealed that Strategy 6 which implements a combination of all optimal control measures is the most effective strategy for minimizing the spread of the coinfection of the epidemics, whereas Strategy 1 which implements rodenticide control measure is the most effective when available resources are limited.

This paper successfully employs a combined methodology that integrates the reduced differential transform approach, Laplace transform, and Padé approximants to solve diverse partial differential equations with real and complex variables. The proposed method, known as the modified reduced differential transform method (MRDTM), extends the interval of convergence with less computing time. An interesting aspect of this method is its capability to produce an analytic exact solution with only a few computable terms. The paper provides practical applications through notable examples encompassing the Klein–Gordon equation, the Schrödinger equation, the nonlinear reaction–diffusion–convection equation, and systems of linear and nonlinear PDEs. Primary results on certain test problems demonstrate the efficiency and ability of the method in solving diverse classes of partial differential equations, and therefore, it can be used as an alternative for dealing with such problems that do not have analytic solutions.

We propose a unified numerical framework for the transport of passive pollutants by shallow-water flows. The mathematical model we consider for describing this phenomenon results in the coupling of the hydrodynamic shallow-water equations with a two-dimensional advection–diffusion equation governing the pollutant transport. The numerical implementation of this hyperbolic model, based on the finite element method, is achieved using a multiphysics modeling and simulation toolbox featured by Feel++, a versatile C++ library applying the Galerkin methods for solving partial differential equations. Numerical experiments targeted on arsenic, cadmium, and lead, heavy metals among the most harmful to human health, are presented as part of a practical application on the Niger River in Bamako. The framework is validated on the basis of RMSE and MAE metrics, some of the most commonly used error measures in linear regression, using observational data. These indicators, estimated below 5% of the observed mean value, support the reliability and accuracy of the numerical model in capturing pollutant dynamics under flow conditions. The simulation results highlight the predictive effectiveness of this framework and provide better insight into pollution patterns in the scrutinized river section.

Despite the rapid growth of enterprises in China, logistics costs remain a significant challenge, accounting for a large proportion of overall business expenses and hindering corporate development. Effective logistics cost management requires careful coordination of routing and resource allocation for goods transportation. Studies have highlighted several challenges faced by China’s logistics industry, including high transportation costs, complex management processes, and excessive warehousing expenses. To tackle these issues, this paper proposes an optimization model for logistics costs utilizing Internet of Things (IoT) technology. By improving the efficiency and accuracy of each stage in the logistics supply chain through advanced information processing, the model seeks to streamline supply chain operations and reduce transfer costs. Additionally, the paper explores the current state of e‐commerce enterprise growth and future logistics control trends, providing strategic recommendations. These strategies include optimizing logistics information flow through IoT, developing IoT‐based cost management systems for e‐commerce logistics, and integrating logistics cost optimization into the strategic planning of e‐commerce enterprises.

This study explores the impact of the Cattaneo–Christov heat and mass fluxes, viscous dissipation, and nonlinear thermal radiation on the stagnation point flow of electrically conductive Casson hybrid nanofluids over a porous material. Employing a similarity transformation, the nonlinear partial differential equations are reduced to nonlinear ordinary differential equations, which are then numerically solved using MATLAB’s bvp4c solver. The results highlight the influence of nanoparticle fraction on velocity, temperature, heat transfer, and drag forces, with hybrid nanofluids outperforming single nanofluids. Increased nanoparticle fraction improves particle interaction and thermal conductivity.

Leptospirosis is a zoonosis with global distribution, and a wide variety of clinical symptoms often lead to misclassification as other febrile conditions. Clinical misclassification has remained the baseline for the diagnosis of leptospirosis, which poses an uphill challenge to clinical management and epidemiological modeling, which could distort the estimation of our burden of disease, thus further delaying public health interventions. This paper provides an overview of trends in modeling approaches for leptospirosis, with a focus on one of the major challenges of diagnostic inaccuracies in relation to effects on model reliability. Finally, the shortcomings of the classic models are discussed in the context that misdiagnosis has not been well represented, and heeding the strides that have recently been made towards developing ways in which diagnostic uncertainty can be incorporated within these frameworks. Enhanced model accuracy of leptospirosis for robustness will help enhance our understanding of the dynamics of diseases to better inform effective intervention strategies. The importance of interdisciplinary communication between epidemiologists, clinicians, and modelers in addressing the misdiagnosis of infectious disease models is outlined herein.

On the global arena, drug and substance abuse (DSA) problem has caused unprecedented suffering. The number of the youth and adult population developing addictions has been on a steady rise, according to the United Nations Office on Drugs and Crime (UNODC) report. A new mathematical model incorporating the impact of policing and rehabilitation on proliferation of DSA in the community is designed and examined in this research. This study postulates that the propagation of DSA is comparable to that of horrendous contagious disease. The effective reproduction number ( R 0 ) of the proposed deterministic model provides the substratum for stability analysis. From the analysis, the DSA‐free equilibrium (DSAFE) is corroborated to be locally asymptotically stable when R 0 < 1 and unstable for R 0 > 1. The local stability of DSA endemic equilibrium (DSAEE) is explored by employing the centre manifold technique. The study rules out the possibility of occurrence of backward bifurcation. Using Python software, various simulation scenarios were run in order to correlate the outcomes with the analytical findings. The simulation results show that increasing the rate at which light drug abusers (LDAs) and heavy drug abusers (HDAs) are identified and placed under structured treatment and therapy, as well as proactive response by intercepting and dismantling trafficking networks via efficacious law enforcement, would adversely diminish drug abuse menace not only in Kenya but in the world at large.

Microfinance institutions (MFIs) play a unique role in the financial sector, using an alternative financial intermediation system (business model) to provide banking services to the marginalized. This is particularly important in areas where collateral-based conventional banking could be more effective. Thus, access to financial services, particularly microfinance loans, is crucial for developing small and medium-sized enterprises (SMEs), especially in rural areas where traditional banking services may be inaccessible. The objectives of this study are to investigate the extent to which factors other than interest rates impact microfinance loan allocation, evaluate the acceptable level of expected loan loss (ELL) that banks can tolerate without compromising financial stability, and explore how banks strategically allocate assets to risky loans under uncertain market conditions. The results from the ELL function indicated that varying risk profiles significantly influenced sensitivity to changes in loan size. This, in turn, affected the institution’s risk sensitivity and tolerance levels at each branch or with each loan product, thereby aiding in the appropriate loan allocation. The recommendations based on the studies include using nonprice terms, loan evaluations, and strengthening branch-level decision-making by empowering branch managers with the necessary tools and training to make decisions that reflect the local context and specific loan products.

This work investigates the concept of numerically approximating fractional differential equations (FDEs) by using function average in an interval. First, the equivalent integral equation is obtained. Then, averaging methods (first-order product integration methods), such as Euler and midpoint, consist in replacing the right-hand side of the FDE by its point evaluation within the interval of integration. In this work, we generalize this idea to leverage on any numerical quadrature rule from classical calculus. Our idea is based on viewing this point evaluation as an average of the function. By definition, this average is a definite integral which can be separately approximated by any classical quadrature rule. A more accurate quadrature rule leads to a more accurate method for the FDE. Here, we use the nonstandard fourth-order trapezoid rule, and this leads to a prediction–correction method. We formulate a modified version of the corrector for the predictor. The convergence of the method is proved. Several numerical experiments are presented to compare the new trapezoid rule with Euler’s and midpoint methods. The results reveal that the new method is more accurate than the Euler and midpoint methods. The conclusion from the work is that a more accurate classical quadrature rule for the averaging leads to a more accurate solution of the FDE.

The study examines the convergence analysis of a higher order approximation for a transport equation with both nonhomogeneous and homogeneous boundary conditions by using the Crank–Nicolson and their modified schemes. Using the Taylor series expressions, the suggested numerical schemes are created. Von Neumann stability analysis combined with the error-boundedness criterion provides a complete examination of stability and convergence. The analysis’s findings show that both Crank–Nicolson systems exhibit unconditional stability and are convergent with second-order accuracy in both directions. Using the current two approaches, extensive numerical experiments are carried out. The outcomes of carried-out trials are compared to the accuracy of the previous schemes in the literature. The comparative analysis demonstrates that the current approaches yield higher accuracy than the earlier techniques. In addition, we contrast the results of the two approaches and compare them. Based on these findings, the accuracy of the modified Crank–Nicolson scheme is better than that of the original Crank–Nicolson scheme. Overall, this research demonstrates that current methods effectively solve partial differential equations.

An essential tool for studying the web is its ability to show how energy moves through an ecosystem. Understanding and elucidating the relationship between species variety and their placement within the inclusive trophic dynamics is also beneficial. A food web ecological model with prey and two rival predators under fear and wind flow conditions is developed in this article. The boundedness and positivity of the system’s solution are established mathematically. The stability and existence constraints of the system’s equilibria are examined. The proposed system’s persistence limitations are established. Additionally, the bifurcation analysis of every potential equilibrium is examined using the Sotomayor theorem. To describe the dynamical behavior of the proposed system, inclusive numerical simulations are performed using MATLAB software (version R2018b). This article aims to recognize the effect of fear and wind flow on the dynamics of this ecosystem. It was found that the high levels of fear caused the decrease of the predators, and the bistable state appeared while rising levels of wind flow caused the extinction of the predators.

We consider the problem of computing the soccer ball trajectory, that is, given the initial velocity, the initial angular velocity, the wind velocity, and the soccer ball geometry, we compute the soccer ball positions at different times during the ball flight. For the solution of this problem, the classical model of projectile motion considering only the gravity force has been equipped with fluid dynamics forces including eventual wind effects. Understanding these forces is essential for improving both gameplay and equipment design. The starting point of the proposed method is a deep revision of fluid dynamics results to provide an effective computational procedure for the evaluation of the drag force and the lift force on the soccer ball. The resulting computational framework is a quite flexible tool that allows the easy management of the main conditions affecting the ball trajectory. So, it can support the training activity and the ball design. Some numerical experiments show the efficacy of this computational tool.

Nosocomial infections, such as those caused by carbapenem-resistant Klebsiella pneumoniae (CRKP) transmitted into hospitals, worsen patient outcomes, strain healthcare infrastructure, increase treatment costs, prolong hospital stays, and complicate medical treatment due to their high level of antibiotic resistance. Taking into account these detrimental aspects, this study deals with a dynamic system of nonlinear differential equations with maintenance preventive strategies, and treatment applying combination therapy has been considered to prevent CRKP transmission and reduce antibiotic resistance while accounting for two hospital populations. The analytical analysis includes positivity, boundedness, existence of a unique solution, stability at equilibrium points, reproduction number (R0), sensitivity, and convergence of state variables. And, the numerical simulations were performed using the fourth-order Runge–Kutta method. We find that the proper implementation of hygiene practices among hospitalized patients and the consistent use of personal protective equipment (PPE) by healthcare workers play an important role in preventing CRKP transmission in hospitals. Additionally, applying combination therapy as a treatment approach for CRKP reduces antibiotic resistance. Consequently, this improves hospital safety and ensures a healthier environment by minimizing the spread of infectious agents and reducing the risks of nosocomial infections.

This research examines the behavior of interfaces in nonlinear multidimensional reaction–diffusion equations with parabolic p-Laplacian properties, which are applicable across a wide range of biological, physical, and chemical contexts. The value of this work lies in its contribution to understanding how interfaces behave under slow diffusion, shedding light on the complex interplay between diffusion and reaction forces. The study aims to analyze the existence and dynamics of interfaces governed by a Cauchy problem, particularly focusing on their expansion, contraction, or stability, influenced by different system parameters. The methodology incorporates the formulation of weak solutions, rescaling techniques, and self-similar solutions to derive detailed expressions for the local interface behavior. The main conclusion is that the behavior of the interface, whether it expands, contracts, or remains stable, is strongly governed by the parameters p, λ, and q. Additionally, the finite propagation speed ensures that the effects are confined, making the model applicable to practical scenarios such as tumor growth, porous media flow, and phase transitions.

In this study, the Bianchi Space Metric-I cosmological model is introduced within the context of the Saez-Ballester theory of gravitation, including the incorporation of scale factors in the framework of gravity. The objective is to derive an exact solution for the cosmological field equation, with consideration given to the Landau and Lifshitz energy tensor. This solution includes a metric potential comprising commoving vectors and an energy conservation equation. Within this framework, a set of relations between the deceleration parameter, Hubble parameter, and average scale factor is established. By adopting a probable set of relations, the influence of the dynamics of the deceleration parameter on energy density and isotropic pressure, particularly in exponential form, is explored. A key proposal involves the utilization of a linearly decelerating parameter alongside exponential scale factors. The results are presented graphically, offering insights into potential future cosmological models. These graphical representations are intended to facilitate the understanding of numerous physical and kinematical properties inherent in the cos-mological model under investigation. Keywords Bianchi Space Metric-I, Saez-Ballester Theory, Landu and Lifshitz Energy Tensor, Bulk Viscosity, Hubble and Deceleration Parameter How to cite this paper: Karim, R.Md. and Islam, M.A. (2025) Bianchi Space-Time Metric -I in Landau and Lifshitz Energy Tensor, Including Linearly Varying Deceleration Parameter with Saez-Ballester Theory of Grav-itation.

We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier-like curves, termed “trigo-curves”, which retain the fundamental properties of classical Bézier curves while offering enhanced shape control through parameter adjustment, independent of control point positions. We establish the monotonicity-preserving nature of these curves. Additionally, we develop a new class of linear positive operators based on these trigonometric basis functions. The operators, incorporating an auxiliary parameter, are thoroughly analyzed for their fundamental properties. We establish their convergence rate, derive a modified Voronovskaja theorem, and obtain error bounds in terms of the modulus of continuity. Furthermore, the monotonicity-preserving properties of these operators are also investigated.

In response to the problems in solving large-scale linear algebraic equations, this study adopts two block discrimination criteria, the Euclidean clustering and the cosine clustering, to decompose them into row vectors and construct the K-means clustering algorithm. Based on the uniformly distributed block Kaczmarz algorithm, the weight coefficients are integrated into the probability criterion to construct a new and efficient weight probability. Thus, a block Kaczmarz algorithm based on weight probability is constructed. Experimental results showed that the method was three to five times faster than the greedy Kaczmarz method. In any convergence situation, the uniformly distributed Kaczmarz method had smaller computational complexity and iteration times compared with the weighted probability block Kaczmarz method, with a minimum acceleration ratio of 2.21 and a maximum acceleration ratio of 8.44. This study can establish and analyze mathematical models of various complex systems, reduce memory consumption and computation time when solving large linear equation systems, and provide solutions and scientific guidance for solving large linear equation systems in fields such as civil engineering and electronic engineering.

This work is aimed at deriving a computationally efficient approach to approximate the second‐order Index‐1 descriptor systems without exploiting the fundamental structure of the systems, which ensures both the accuracy of the approximation and the feasibility of computation time. It demonstrates the structure‐preserving iterative singular‐value decomposition (SVD)‐Krylov algorithm (ISKA) which is a hybrid two‐sided projections strategy combined with a computationally feasible Krylov subspace technique and the stability‐preserving iterative technique based SVD. ISKA is a model‐order reduction (MOR) approach utilized to get a reduced‐order model (ROM) corresponding to a target model with a desired size, where two projector matrices are the main constituent. The left projector matrix will be constructed using the low‐rank Cholesky‐factor alternative direction implicit (LRCF‐ADI) technique, whereas the right projector matrix will be constructed utilizing the well‐known Krylov subspace. The main focus of this work is to avoid the so‐called first‐order conversion of the second‐order systems and ensure the invariance of the matrix‐vector operations of the original systems. A real‐world model derived from the adaptive spindle support system will be utilized as the target model to numerically validate the competency and efficiency of the proposed strategy through MATLAB simulations. Accuracy of the approximation, stability of the ROMs, and optimization of computational time are the prime concerns of this work. To confirm the advancement of the proposed approach, it will be compared with the existing approaches employing the necessary figures and tables.

Considering the importance of ecosystems from an economic and biological point of view, a new ecological model consisting of prey and a predator with an infectious disease in the presence of fear of predation and refuge has been proposed and studied. The research is aimed at studying these factors’ influence on the predator–prey system’s dynamic behavior in the presence of infectious diseases. The system’s solution properties and limitations were studied. All the necessary conditions for the system’s stability locally and globally are calculated using various mathematical methods. The possibility of local bifurcation in the system has been studied. Finally, the system was solved numerically to confirm the results and understand the effect of changing parameter values on dynamic behavior. It has been found that, up to a critical value, the system is stabilized by the fear of susceptible predators; beyond that, predator extinction threatens the system. A similar observation has been made regarding the refuge rate. Nonetheless, the system dynamics are destabilized by the fear of the infected predators.

In this study, an optimal control problem is formulated to a prey–predator model with disease in both species. The model is an adapted Lotka–Volterra model by incorporating susceptible–infected (SI) epidemic dynamics on the prey–predator population. A sensitivity analysis is conducted. The result of the analysis shows that the infection rates β1 and β2 and the conversion rate e1 are critical, influencing the dynamics through interaction and efficiency. The time-dependent control is implemented in the system to determine the most effective disease control strategy. The optimal control is characterized using Pontryagin’s optimality principle, which involves introducing the Hamiltonian and its associated adjoint variables. The study found that separating infected populations is crucial for disease elimination. We examine numerical simulations of various system scenarios using a wide range of parameters. Simulations are performed using a forward–backward sweep method with first-order necessary conditions for the control problem. The optimal solution is determined through numerical methods. The findings indicate that the ideal combination of the three control strategies necessary to meet the specified objective is influenced by the relative costs associated with each control measure. A detailed discussion of the simulation results is provided.

Excessive intake or injection of substances, namely, medications, alcohol, and other harmful drugs, has resulted into unimaginable serious consequences, including mental health and social problems. In an attempt to understand the dynamics of substance abuse and forestall its potential spread in the population, a novel model based on nonlinear system of ordinary differential equations is formulated and analysed in this study. The model takes into account, among other important features, the influx of addicted immigrant and rehabilitation of individuals affected by substance abuse. Least squares method with minimization-constrained function is employed to fit the model with the real data of substance-induced mental cases under rehabilitation. Conditions that guarantee the existence and global asymptotic stability of steady states are established, and a key threshold quantity which measures the potential spread of substance abuse influence in a community comprising susceptible and prudent populations is determined. Sensitive parameters of the model are identified, and their effects on the dynamics of substance abuse transmission are investigated with a view to suggesting possible effective measures against the harmful spread of substance abuse in the population.

In this article, a fractional-order mathematical model is used to simulate peer influence using the Liouville–Caputo framework. Our model was made up of four states, which describe friends, negatively behaved friends, parental guidance, and positively behaved friends. We found the equilibrium points and also did the stability analysis to ascertain the conditions necessary for a stable solution. Again, we established the uniqueness, existence, and boundedness of our solution through the use of the Banach fixed point theorem and also checked for the global stability of our equilibrium points using the Lyapunov function. We finally conducted a numerical simulation with various parameters and fractional orders, demonstrating the effectiveness of our method. Our study revealed that the various fractional orders used have a great impact on the behavior of the model. In addition, we found that negatively behaved individuals have a greater influence on other individuals, so for us to curb or lower their associations and interactions, parental guidance must be intentionally increased. Our study contributes to the understanding and dynamics of peer influence through mathematical modeling.

This article extends a spectral collocation approach based on Lucas polynomials to numerically solve the integrodifferential equations of both Volterra and Fredholm types for multi–higher fractional order in the Caputo sense under the mixed conditions. The new approach focusses on using a matrix strategy to convert the supplied equation with conditions into an algebraic linear system of equations with unknown Lucas coefficients. The coefficients of the presumed solution are determined by the solution of this system. The Lucas coefficients are used to track how the solutions behave. This method is attractive for computation, and usage examples and explanations are provided. Additionally, certain examples are provided to demonstrate the method’s accuracy, and the least-squares error technique is employed to reduce error terms inside the designated domain. Because of this, Python is used to write most general programs.

This study addresses the pressing challenge of mitigating bed bug infestations in urban and residential settings under the constraints of limited extermination resources. To this end, we develop and analyze a susceptible–infectious–treatment–reservoir (SIT-R) epidemiological model that captures the interactions between humans, domestic animals, and bed bugs residing in the environment. The model provides a robust framework for understanding the dynamics of common bed bug infestations and evaluating control strategies. Using the next-generation matrix approach, we derive the basic reproduction number, ℝ0, as a critical threshold parameter. We establish that the infestation-free equilibrium is locally stable when ℝ0<1, highlighting the conditions necessary for the successful control and eventual elimination of infestations. To complement the theoretical findings, we perform numerical simulations, which illustrate key properties of the model. Our results underscore the importance of targeted disinfestation strategies, particularly those aimed at reducing the bed bug population in environmental reservoirs. Even when resources for extermination, such as insecticides and personnel, are limited, prioritizing interventions in the reservoir proves to be highly effective in lowering infestation rates and preventing the spread of bed bugs. These findings provide actionable insights for public health officials and pest management professionals, offering a pathway toward more efficient and sustainable bed bug control measures.