In this paper, we completely solve the Diophantine equations Pm=Fn+Fk and Em=Fn+Fk under the condition n≥k+2≥4, where Pm is the m-th Padovan number, Em is the m-th Perrin number, and Fk is the k-th Fibonacci number. This work determines all terms of the Padovan and Perrin sequences that admit a unique two-term Zeckendorf representation. Our approach relies on lower bounds for linear forms in logarithms of algebraic numbers via Baker's method, combined with the Baker–Davenport reduction procedure to reduce the large initial upper bounds. We show that the only Padovan numbers satisfying this property are 4,7,9,16,37, corresponding to the indices m∈7,9,10,12,15, and the only Perrin numbers are 7,10,22,29,39,68,90,644, corresponding to the indices m∈7,8,11,12,13,15,16,23.
The work is aimed at proposing a novel iterative technique based on the combination of two powerful Taylor's and Bernoulli polynomials to solve nonlinear Volterra–Fredholm integral equations of the 2nd kind. The main contribution of this work is the development of a Bernoulli–Taylor iterative framework in which Bernoulli projection is combined with Taylor approximation within the nonlinear operator, leading to an explicit error estimate that simultaneously accounts for the Bernoulli approximation, Taylor truncation, and iterative errors. For this aim, an approximation of the unknown function by the Bernoulli polynomial with different degrees is proposed, and then the solution is obtained by taking the limit of the resulting solutions. The Taylor series expansion is taken for the other known functions in the integral equation. Also, theoretical analysis establishes the convergence and provides a general error estimate for the proposed method under the stated assumptions, whereas the numerical examples confirm its accuracy and effectiveness for the considered test problems, which illustrates that the proposed approach is valid not only for Volterra–Fredholm problems but also for Volterra and Fredholm nonlinear integral equations, and in each case, the solution converges to the exact solution. Finally, the numerical results are further validated through comparisons with existing methods reported in the literature, as well as by evaluating the maximum absolute error, demonstrating the high accuracy and effectiveness of the proposed method.
Fractional differential equations (FDEs) are of great significance for describing complex physical processes with memory and genetic characteristics. The introduction of the p ‐Laplacian operator can expand its applicability in nonlinear problems. To explore the uniqueness multiplicity, and variety of solutions to p ‐Laplacian differential equations with singular terms, the influences of fractional order and P‐Laplace exponent on the solution morphology are analyzed. Based on the Riemann–Liouville theory, the boundary value problem is transformed into an equivalent integral equation through integral transformation and the Green′s function is constructed. The existence theory of the solution is established, and then the influence of parameters is verified and analyzed through numerical examples. The results show that when the p ‐Laplacian index is 1.8, 2.5, and 3.2, the peak solutions are 0.7768, 0.6393, and 0.5229, respectively. The results show that when a specific integral inequality is satisfied reducing the p ‐Laplacian exponent increases the amplitude of the solution. Under appropriate nonlinear conditions, the equation has three positive solutions (PS) with large amplitude differences. This research provides methodological references and theoretical foundations for the analysis of related nonlinear FDEs.
Information dissemination and underlying social structures profoundly influence human behavior and epidemic dynamics. This study introduces a compartmental framework, the SVAIRDM (susceptible–vaccinated–isolated–infected–recovered–deceased–information) model, which explicitly incorporates an information index to quantify the effects of public health information, including misinformation, on individual behavioral responses. The model is embedded within a two‐layer network architecture composed of Barabási–Albert (BA) scale‐free networks and Erdős–Rényi (ER) random networks, both of which capture the coevolution of disease transmission and information spread, reflecting the complex interplay observed in real‐world epidemics. Beyond epidemic dynamics, we examine the social dilemma associated with individual decision‐making in vaccination and isolation through the social efficiency deficit (SED) lens. Our analysis highlights how information‐driven behaviors shape collective welfare by quantifying the gap between socially optimal and individually rational outcomes. Interestingly, we observe instances of negative SED under certain conditions of the information index, suggesting that, paradoxically, intensified information dissemination may worsen collective outcomes despite improving individual‐level strategies. Our results show that the information index enhances epidemic control by facilitating earlier isolation and higher vaccination rates, with BA networks enabling rapid behavioral adaptation through highly connected hubs. In contrast, ER networks produce steadier, more uniform behavioral shifts. These findings underscore the importance of incorporating both behavioral–social dilemmas and information‐driven dynamics into epidemic modeling. This work provides a comprehensive framework for developing communication strategies that support vaccination and isolation policies and address the inefficiencies arising from social dilemmas, thereby offering more profound insights into optimizing epidemic mitigation in complex and densely connected populations.
Trachoma remains a leading cause of preventable blindness in many low‐resource settings, driven by persistent transmission within closely interacting age groups. In this study, we develop and analyze a nonlinear age‐structured compartmental model that captures the transmission dynamics of trachoma between children and adults. The model incorporates cross‐age infection pathways, demographic processes, and Holling Type‐II saturated incidence to reflect behavioral and environmental limitations on disease spread. We establish fundamental qualitative properties of the model, including positivity, boundedness, and well‐posedness of solutions. Using the next‐generation matrix approach, the basic reproduction number is derived and shown to govern the global dynamics of the system. Specifically, the disease‐free equilibrium is globally asymptotically stable when , whereas a unique endemic equilibrium exists and is globally asymptotically stable when . Furthermore, bifurcation analysis reveals the presence of a forward (transcritical) bifurcation at , indicating that disease elimination can be achieved through effective threshold reduction strategies. A comprehensive sensitivity analysis, based on Latin Hypercube Sampling and Partial Rank Correlation Coefficients (LHS‐PRCC), identifies transmission parameters—particularly child‐to‐child and adult‐to‐child contact rates—as the most influential drivers of disease spread, whereas recovery and mortality parameters play a significant mitigating role. Numerical simulations are conducted to evaluate the impact of intervention strategies. Results demonstrate that biannual mass drug administration (MDA) targeting children substantially reduces short‐term prevalence but fails to prevent long‐term resurgence when implemented in isolation. In contrast, adult‐targeted hygiene education reduces transmission intensity and lowers endemic levels. A combined intervention strategy yields a strong synergistic effect, significantly accelerating disease suppression and reducing long‐term prevalence. Taken together, the findings highlight the critical role of integrated control strategies that combine medical treatment with behavioral interventions. In particular, targeting adult‐mediated transmission pathways is essential for sustainable trachoma elimination and aligns with the WHO SAFE strategy.
In this paper, we define the complex dual fuzzy matrix equation and solve it based on the study of the dual core-EP inverse. First, we define the complex dual fuzzy matrix equation. Second, by characterizing some properties of the dual core-EP inverse, we get the block structure of the dual core-EP inverse corresponding to the coefficient matrix of the fuzzy complex number matrix equation that is then used to obtain a strong fuzzy solution of the complex dual fuzzy matrix equation. Third, we derive the general strong fuzzy solution of the complex dual fuzzy matrix equation based on the dual core-EP inverse or a strong fuzzy solution and establish an algorithm to solve this equation by using the dual core-EP inverse. Lastly, several numerical examples are provided to demonstrate the main results.
The aim of the work is to investigate a nonlinear reaction–diffusion model applied to electroactive polymer films, formulated within a fractional-order framework to incorporate spatial nonlocality and anomalous transport effects. The conventional model is generalized using the Liouville–Caputo fractional derivative to provide a more realistic description of anomalous diffusion. The resulting strongly nonlinear fractional boundary value problem is solved using the Differential Transform Method (DTM), which yields accurate semianalytical approximate solutions. In addition, a detailed perturbation-based stability analysis is performed, leading to an explicit expression for the stability coefficient. The results of the stability analysis clearly show that the system is inherently unstable, and this instability is further amplified by the spatially nonlocal transport effects introduced by the fractional operator. The impact of the fractional order is also investigated, showing that decreasing the fractional order enhances spatial nonlocal interactions and anomalous transport, thereby accelerating the growth of perturbations, whereas the integer-order limit gradually recovers the classical diffusion behavior. The numerical and graphical results are in good agreement with the previously reported results, validate the analytical results, and show the effect of the governing parameters. The proposed analytical framework offers a deeper understanding of the interplay between nonlinearity, saturation, and spatial nonlocal transport in fractional reaction–diffusion systems and provides an efficient methodology for the analysis of complex transport phenomena in electroactive polymer films.
Substance misuse is a serious threat to people's health and the socioeconomic stability of countries all over the world. Tobacco and marijuana are among the most abused drugs. The joint misuse of the two substances has made the drug problem in the community worse. In this study, a mathematical model was developed to explain the dynamics of the coabuse of marijuana and tobacco, as well as the impact of an educational campaign. In this research, the population is divided into seven compartments: the susceptible population (S), light smokers (L), heavy tobacco smokers (T), marijuana smokers (M), both tobacco and marijuana smokers (B), individuals undergoing rehabilitation (R), and the quitting population (Q). In addition, the study incorporated an educational campaign to determine its implications on those addicted to tobacco and marijuana use. In the study, the presence of equilibrium points, the endemic equilibrium, and the drug-free equilibrium of coabuse were analyzed. The next-generation matrix method was used to determine the basic reproduction number. The model was numerically simulated using Maple, which employs the Runge–Kutta fourth-order method for interpretation and qualitative analysis. From the numerical analysis of the coabuse model, the results showed that educational campaigns influence addicted individuals to go for treatment. This suggests that there is a need to intensify and improve the available educational campaigns to help reduce the menace in society.
This study introduces a fully algorithmic framework for analyzing the dynamics of an MSEIR (Maternal, Susceptible, Exposed, Infectious, Recovered) epidemic model. Unlike traditional approaches that rely on heuristic, model-specific Lyapunov functions for global stability or manual algebraic manipulations for finding equilibria, we employ computational algebraic geometry. By computing Gröbner bases, the coupled nonlinear ordinary differential equations are systematically reduced to a triangular form, guaranteeing the exhaustive determination of all equilibria. Local stability is rigorously established via the Liénard–Chipart criterion applied to the Jacobian's characteristic polynomial, bypassing explicit eigenvalue extraction. Furthermore, we demonstrate a transcritical bifurcation where the disease-free and endemic equilibrium exchange stability. Finally, the global asymptotic stability of the disease-free equilibrium for R0<1 is proven using an algebraic matrix approach, offering a reproducible and extensible pipeline for compartmental epidemiological models.
This paper presents a mathematical analysis of a nonautonomous delay differential system. This system models mosquito population dynamics under the influence of seasonal environmental fluctuations and a biological control strategy based on the continuous release of Wolbachia-infected males. The model extends previous frameworks by incorporating a maturation delay for aquatic stages and time-periodic coefficients for all demographic parameters, thereby capturing two essential biological features: the developmental lag between aquatic and adult stages and the seasonal variability in fecundity, carrying capacity, and mortality rates. We establish the well-posedness of the system by proving existence, uniqueness, positivity, and boundedness of solutions. Using the theory of semiflows and comparison principles for functional differential equations, we derive the basic offspring number Rv in a periodic environment with delay via the next-generation operator approach. A sharp threshold dynamics is demonstrated: When Rv<1, the trivial periodic solution corresponding to population extinction is globally asymptotically stable, whereas when Rv>1, the system exhibits uniform persistence and admits at least one positive periodic solution. Extensive numerical simulations illustrate the theoretical findings and investigate the sensitivity of population dynamics to key parameters, including the maturation delay and the amplitude of seasonal variations in oviposition rate and carrying capacity.
This study investigates the chiral nonlinear Schrödinger equation for current-dependent nonlinear wave propagation in the fractional quantum Hall setting when temporal evolution is represented by the conformable fractional derivative. This derivative is defined operationally as the ordinary first time derivative multiplied by a time-dependent power factor determined by the fractional order. The physical objective is to clarify how fractionally rescaled temporal evolution modifies coherent chiral excitations associated with quantum Hall edge-state dynamics. The novelty lies in a unified quartic auxiliary-equation classification based on the modified sub–ODE method rather than the derivation of isolated wave forms. A conformable traveling-wave reduction produces an amplitude problem whose admissible branches yield localized solitary waves, kink-type waves, singular structures, Jacobi elliptic waves, Weierstrass elliptic waves, and mixed solution families with explicit existence restrictions. The results are validated by direct substitution into the reduced and original models, together with first-integral and limiting-case checks. The analysis shows that, for fixed reduced coefficients, the fractional order changes the temporal scaling, phase evolution, and propagation trajectory while preserving the analytical amplitude family. This separation provides a useful mechanism for tuning the phase and propagation of chiral wave structures without altering their intrinsic analytical profile.
Partial differential equations serve as essential tools for describing the evolution of complex systems in physics, engineering, and other fields. Traditional modeling methods often encounter difficulties such as local optima in structure search and weak noise resistance in complex scenarios. To address these issues, this paper proposes an efficient discovery method for partial differential equations by integrating reinforcement learning, genetic algorithm, sparse identification of nonlinear dynamics, and Bayesian optimization. The method first performs a global evolutionary search of candidate structures for partial differential equations through an improved genetic algorithm. Then, it applies a sparse identification algorithm to remove redundant terms by sparse regression. Finally, it uses a Bayesian optimization to optimize key hyperparameters such as regularization strength, crossover probability, and mutation probability, while quantifying uncertainty. Experimental results show that the proposed method performs well on the Tsinghua partial differential equation operator large model dataset and the fluid mechanics dataset. The structure accuracy of the discovered equations reaches 98.32%, and the training time for discovering equations is only 172.7 ms (excluding the time needed to train RL model), which is much faster than other compared algorithms. These results demonstrate that the collaboration among multiple algorithms enables accurate and efficient discovery of partial differential equations under complex environments, providing technical support for system modeling and rule mining in multiple fields.
This study extends asymmetric analysis by introducing an asymmetric wavelet quantile regression (AWQR) framework to examine how positive and negative changes in US–China tensions (UCTs) and oil prices (OPs) affect the global clean energy transition (GCET). Using monthly data from 01/03/2007, to 01/02/2024, the framework captures distributional heterogeneity, frequency heterogeneity, and sign asymmetry. The findings show that symmetric and asymmetric specifications provide distinct insights. Under the symmetric model, UCT has little immediate effect on GCET, but its influence becomes stronger over the medium and long terms, with negative effects at upper long-term quantiles. Positive UCT shocks generally weaken GCET at middle medium-term quantiles but support it at upper long-term quantiles, whereas negative shocks improve GCET at selected medium-term quantiles but reduce it across middle and upper long-term quantiles. OPs generally promote GCET, especially in the medium term. Positive OP shocks strengthen GCET mainly at middle and upper quantiles, while negative OP shocks support GCET in the medium term but weaken it at upper long-term quantiles. The study proposes policy recommendations based on these findings.
Currently, ordinary differential equations play an important role in various fields, which is of great significance for scientific research. However, traditional inversion methods for ordinary differential equations often suffer from poor inversion bias and poor ability to optimize equation parameters. To improve the inversion performance of ordinary differential equations, a linear and nonlinear equation inversion method based on an improved multistep neural network (MSNN) is proposed. A new method for ordinary differential equation inversion based on improved MSNNs and a two-stage Monte–Carlo optimization is proposed. An MSNN architecture with a time-step adaptive mechanism is designed to uniformly handle linear and nonlinear problems, and a two-stage optimization strategy combining coarse sampling and covariance matrix fine-tuning significantly improves parameter estimation efficiency. L2 regularization and momentum sampling are introduced to enhance noise resistance. Experiments showed that this method reduced the maximum inversion deviation by 3.2 and shortened the convergence time by 24 s, providing an efficient and stable solution for complex system parameter inversion.
Malaria remains a major vector-borne infectious disease, particularly in tropical and subtropical places, where Anopheles mosquito bites are the primary means of transmission. Human mobility plays a crucial role in influencing the geographical distribution of malaria, whereas temperature variations significantly affect mosquito biting behavior and mortality rates. To control malaria transmission across two patches with varying degrees of endemicity, we proposed and analyzed a two-patch compartmental model that includes patch-specific optimal controls, such as treated mosquito nets, insecticide spraying, and antimalaria drugs. We first established that the model solutions remain bounded and nonnegative, thereby confirming the mathematical and epidemiological validity of the model. Using the next-generation matrix technique, we computed the basic reproduction number, R0m. It is critically dependent on both temperature variability and human movement. Stability analysis indicates that the malaria-free equilibrium point is both locally and globally asymptotically stable when R0m<1 and becomes unstable if it exceeds one. Actual prevalence data for malaria in Ilu Ababor and Gambella, Ethiopia, from 2018 to 2025 were used to validate the model. To assess the effects of patch-specific interventions, simulations were conducted within an optimal control framework. The results demonstrate that the simultaneous application of all three management strategies effectively reduces the spread of malaria in both patches. The main novelty is the integration of optimal controls across two patches, along with real-data validation and combined patch-specific controls, providing a more realistic and policy-relevant framework than traditional malaria models that focus on a single homogeneous population.
Banks are increasingly exposed to liquidity–risk contagion through interbank refinancing links, confidence effects, and balance sheet interactions. This paper develops a fractional-order extension of a SIR-type liquidity contagion model previously introduced in Mourad et al. (2022). The main purpose is not merely to replace the classical derivative by a fractional one, but to provide a memory-dependent framework in which current banking vulnerability depends on past liquidity shortages, delayed refinancing reactions, and persistent confidence shocks. The model is formulated by means of the Caputo derivative, which preserves classical initial conditions and reduces to the integer-order model when the fractional order is equal to one. We correct the initial shock formulation by allowing a positive initial number of distressed banks, establish well-posedness, positivity, and boundedness, and analyze the equilibrium manifold through a basic contagion threshold. A Lyapunov comparison argument is used to justify the Mittag–Leffler decay of the distressed compartment under the subthreshold condition. An explicit Adams–Bashforth–Moulton predictor–corrector method for Caputo systems is presented and used to produce numerical simulations calibrated from European banking data. The simulations compare several bankruptcy rates, countries, and fractional orders, and they show that stronger memory slows the adjustment process and prolongs liquidity distress.
This study analyzes and forecasts the export value of Vietnamese pepper to the United States by integrating traditional time series models, namely SARIMA-GARCH, with a deep learning approach based on the gated recurrent unit (GRU). The dataset consists of monthly time series observations spanning from January 2002 to December 2025, which are utilized to examine trends, seasonality, volatility, and nonlinear characteristics of export values. The SARIMA-GARCH framework is employed to model both the conditional mean and variance structures, whereas the GRU model is designed to capture nonlinear relationships and long-term dependencies inherent in the data. Empirical results from out-of-sample forecasts indicate that both models perform effectively; however, the GRU model demonstrates superior predictive accuracy in terms of MAE, RMSE, MAPE, and the coefficient of determination (R2). Furthermore, the Diebold–Mariano test and robustness analysis provide additional evidence supporting the stability and generalization capability of the GRU model. This study offers empirical insights into the effectiveness of combining time series and deep learning approaches in forecasting agricultural exports. It also provides valuable implications for policymakers and businesses in formulating sustainable development strategies for Vietnam's pepper industry.
Traffic congestion in Lusaka City has reached critical levels due to rapid urbanization, rising vehicle ownership, and inadequate transport infrastructure. Traditional planning approaches have failed to capture the complex dynamics of traffic flow, highlighting the need for mathematical modeling. This study applies and validates an integrated macro–micro traffic flow model for Lusaka using survey data from 40 drivers, Global Positioning System (GPS) tracking, and traffic volume counts. The model combines macroscopic relationships of flow, speed, and density with microscopic driver behavior representations. Validation against observed data yielded a strong correlation (r=0.759), demonstrating superior predictive performance compared to standalone macroscopic models. Results indicate that congestion is primarily caused by poor road infrastructure, ineffective traffic management, inadequate public transportation, and the high number of vehicles on the road. Policy simulations revealed that 87% of drivers recommended constructing new roads and bridges or expanding existing ones, 5% proposed banning private vehicles, 5% suggested having high levels of car ownership, and 12.5% proposed nonmotorized alternatives. The study's novelty lies in integrating quantitative driver perceptions with advanced traffic modeling tailored to Lusaka's context. Practically, the model serves as a decision-support tool for policymakers to test congestion reduction strategies under real-world constraints.
Beyond human immunodeficiency virus (HIV), silicosis is recognized as a major risk factor for tuberculosis (TB), particularly among mine workers exposed to respirable crystalline silica. Silicosis is an incurable pneumoconiosis caused by the inhalation and deposition of fine silica particles in the lungs. In this study, we develop, for the first time, a TB–silicosis coinfection model to assess the impact of silica dust inhalation on TB transmission dynamics. Without accounting for silicosis infection, we determine a threshold that ensures the global elimination of TB. The coinfection model inherits the backward bifurcation property of the TB-only submodel, in the sense that when the basic reproduction number R0 is less than one, a stable endemic equilibrium coexists with the disease-free equilibrium (DFE), thereby complicating TB elimination efforts. Sensitivity analysis further suggests that, regardless of vaccination coverage or efficacy, sustained silica exposure will maintain TB persistence and even preclude local disease control. Numerical simulations are carried out using a nonstandard finite difference (NSFD) scheme, which is dynamically consistent with respect to the qualitative properties, including the positivity, boundedness, and stability of the DFE of the continuous model. The obtained results indicate that reducing the silicosis progression rate mitigates TB prevalence as well. Additionally, mining recruitment and silica dust shed in the air are important drivers of the coinfection dynamics of TB–silicosis, recommending the implementation of interventions aimed at reducing human exposure to high-risk mining environments (e.g., drilling, blasting, and crushing) through mechanization and automation. Conversely, enhancing silica dust removal significantly reduces the prevalence of silicosis and TB–silicosis coinfection, underscoring the critical role of effective dust control strategies, including the implementation of improved ventilation systems and wet dust suppression techniques to limit exposure to respirable silica dust, thereby lowering the risk of silicosis and contributing to TB control.
In this sequel, we employ an approach that combines the modified version of the Lagrange polynomial approximation technique with the biconjugate gradient stabilized method (BiCGSTAB) to analyze numerical solutions of fractional Volterra integral equations (FVIEs). These equations emerge as indispensable tools in modern physics, enabling the modeling and analysis of intricate dynamical systems with memory effects and nonlocal behaviors. From fractal dynamics to quantum mechanics, electromagnetism, and biophysics, these equations provide a unified framework to capture the rich complexity inherent in diverse physical phenomena. The scheme for establishing the uniquely existence of a solution to the integral equation is elucidated in relation to the Banach contraction principle and the Bielecki norm. Furthermore, we rigorously prove several theorems concerning the method's convergence and error estimation. Notably, our proposed approach effectively handles potential singularities in the solution, a facet that, to the best of our knowledge, has remained unaddressed in existing literature. To showcase the reliability and efficacy of our method, we present illustrative examples along with comparative analyses. All numerical computations are conducted using MATLAB 23.2.0.2410171 (R2023b).
