The sensitivity of chaotic dynamical systems to the initial conditions makes the long-term numerical simulations of such systems challenging since the discretization error can either dampen true chaos or produce artificial chaos. In this paper, we examine the capability of the conventional, advanced, and trigonometric polynomial-based methods in preserving chaos within oscillatory nonlinear systems during lengthy simulation periods. The research procedure is divided into three parts: selecting chaotic systems and numerical techniques, conducting lengthy numerical simulations, and analyzing results based on phase spaces, Lyapunov exponents, Kolmogorov–Sinai entropy, and computation cost. Numerical experiments have been done on the Lorenz, Rossler, Chen, and Chua systems over 50,000s time intervals. It is shown that the backward Euler method fails to preserve the real dynamics in chaotic systems if the appropriate integration steps are not chosen, whereas the forward Euler method creates artificial chaos under instability conditions of the discretization process. Among all numerical methods, the Gautschi scheme offers the best computational efficiency, requiring only 2.7% of the number of computations needed by AB–AM in the case of the Lorenz system but still keeping positive Lyapunov exponents and K-S entropy.
Respiratory infectious diseases that affect the respiratory system continue to pose a serious global threat to public health. Influenza and severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are viral infections that affect the respiratory system and cause respiratory complications and poor lung function. Co-infection occurs because of the contagious nature of both diseases, which exacerbates their severity. The novelty of this study lies in the use of fuzzy parameters and the ABC fractional differentiation to model the respiratory co-infection dynamics caused by SARS-CoV-2 and influenza. Fuzzy parameters were used to represent imprecise and uncertain system quantities more realistically. This approach demonstrated that combining fractional derivatives with fuzzy parameterization yielded stronger and more flexible representations of system dynamics, facilitating a more precise understanding of behavior under uncertainty. The basic reproduction number was determined. For respiratory co-infections caused by coronavirus disease 2019 (COVID-19) and influenza, the crisp model yielded reproduction numbers of 1.779898 and 0.004849, respectively, whereas the trapezoidal fuzzy model yielded defuzzified values of 1.779867 and 0.004849. A low level of uncertainty regarding the estimated transmission dynamics was indicated by the average values of 1.779867 (SD = 0.000458) and 0.004849 (SD = 0.000001) obtained from the fuzzy reproduction numbers. The provided fuzzy model used trapezoidal fuzzy parameters to account for uncertainty while reproducing values close to those of the crisp model. The resilience of transmission dynamics to parameter uncertainty was demonstrated by the low variance of the fuzzy reproduction numbers. The qualitative behavior of the modified model was studied using stability analysis at the equilibrium point. Fractional differential equations of the model based on the ABC derivative were solved numerically using the Laplace Adomian Decomposition Method (LADM), and a graphical representation of the model variables at various fractional powers was obtained for comparison.
We develop a non-linear delay-differential model for within-host HIV dynamics to investigate how biological and pharmacological delays influence viral persistence, immune control, and post-control infection resurgence. The model describes interactions among susceptible CD4+ T cells, productively infected cells, cytotoxic T lymphocytes, free virus, and a protective-response variable, with separate intracellular infection and immune-activation delays. Survival factors are incorporated into the delayed infection and immune-response terms to account for loss during the waiting periods, while the protective response reduces effective susceptibility rather than directly removing susceptible cells. Threshold quantities are derived for infection invasion and CTL activation, and local stability analysis identifies conditions for infection-free, CTL-free endemic, and CTL-activated endemic equilibria. Hopf bifurcation analysis shows how increasing biological delays can destabilize endemic equilibria and generate delay-dependent oscillatory infection dynamics under delayed feedback. Numerical simulations illustrate the transition from stable convergence to transient and sustained delay-dependent oscillations under increasing delays, and global sensitivity analysis identifies the parameters most strongly associated with peak infected-cell burden. We further formulate a delayed optimal-control problem incorporating treatment and prevention controls, together with robustness and post-control resurgence measures. The results highlight the importance of distinguishing intracellular, immune-activation, and treatment-response delays when evaluating HIV suppression, infection-resurgence risk, and intervention timing.
We study the problem of multivariate time-series forecasting, where existing Transformer-based models often treat time-domain dynamics, frequency-domain structure, and cross-channel relationships as separate components, limiting effective multi-view coordination and causing susceptibility to redundant information and spurious correlations. To address this, we propose HMV-Former, a forecasting architecture that integrates multi-view hierarchical patching with lag-coupled channel attention. Specifically, we construct a Hierarchical Patching Stack to build a semantic feature pyramid across multiple scales, design Lag-Coupled Channel Attention to dynamically disentangle asynchronous lagged dependencies from time-varying coupling relations, and adopt an integrated multi-view encoder with attention-logit regularization to constrain information pathways and suppress spurious dependencies in high-dimensional spaces. Experiments on 10 real-world datasets demonstrate that HMV-Former delivers substantial improvements in forecasting accuracy and stability compared to existing methods. These results highlight its advantages in multi-scale feature abstraction and dynamic dependency governance, and suggest its potential to support intelligent decision-making in complex systems. Code and datasets are publicly available at: https://github.com/FangXinbang/HMV-Former.
Educational crowdfunding has emerged as a promising approach to provide educational resources to underprivileged communities. Conventional systems often suffer from a lack of transparency, weak accountability, inefficient allocation of funds, and inadequate traceability of resource use. To address these issues, the present study proposes an intelligent and efficient educational supply chain management system, “EduDonateBlock.” It uses a blockchain-based crowdfunding framework to ensure transparency, accountability, and efficiency. Decentralization, immutability, and verifiable transactions are supported in educational campaigns. The entire workflow is decomposed into modular smart contracts. These are the identity and access contract (IAC), campaign and donation contract (CDC), verification and allocation contract (VAC), and supply chain and tracking contract (SCTC). These contracts are designed to ensure traceability, accountability, and efficient resource allocation among donors, educational institutions, and administrators. The mathematical framework of EduDonateBlock determines the optimal level of blockchain transparency. This minimizes the Total Expected Cost (TEC) of smart-contract operations. Numerical analysis identifies an optimal transparency level of 87.16% on-chain integration. This finding underscores the economic trade-off between transaction costs and the benefits of automation, operational efficiency, and reduced fraud risk. The proposed framework achieves a campaign success probability of 89.45% and an institutional payoff of Rs. 11,335.99. Furthermore, executing smart contracts requires 0.0044 ETH, and the average latency remains at 6.25 s. The simulation results show that EduDonateBlock offers a more efficient, reliable, and transparent solution for decentralized educational crowdfunding and socially impactful digital supply chains.
IntroductionExchange rate volatility in partially dollarized emerging economies poses persistent challenges for financial risk management and monetary policy. Short-memory GARCH specifications provide inferior fit relative to long-memory alternatives under managed floating regimes with recurrent political disruptions.MethodsThis study proposes a two-stage framework: a wavelet-dummy operator maps the return series into a binary indicator of atypical observations incorporated as an exogenous regressor in an ARIMAX(2,0) model; filtered residuals serve as input for five conditional variance specifications-ARCH(1), GARCH(1,1), GJR-GARCH(1,1,1), EGARCH(1,1,1), and FIGARCH(1,d,1)-applied to 2,803 daily log-returns of the USD/PEN (January 2015-March 2026).ResultsThe operator identifies 36 atypical observations concentrated in the 2020-2023 sub-period. FIGARCH(1,d,1) dominates under the Akaike Information Criterion with d = 0.553 (p < 0.001) under Skewed Student-t innovations, extending long-memory evidence for USD/PEN conditional variance to 2015-2026 and resolving prior ambiguity in the Peruvian Forex literature. Ljung-Box tests confirm white noise in both standardized residuals (p = 0.4846) and squared residuals (p = 0.9921). Predictive evaluation across three validation windows yields Theil's U between 0.93 and 1.02, indicating mean-forecast accuracy comparable to the random-walk benchmark, while expanding-window volatility forecasts consistently outperform the constant-variance benchmark.DiscussionThese findings demonstrate that wavelet-based outlier detection combined with long-memory GARCH estimation substantially improves volatility modeling for emerging-market currencies under managed floating regimes, institutional fragility and partial dollarization.
IntroductionCassava is a key staple crop in many developing countries but is severely affected by African cassava mosaic virus (ACMV) and East African cassava mosaic Cameroon virus (EACMCMV). Co-infection by these viruses often results in more severe yield losses than single infections. This study develops a mathematical model to investigate the transmission dynamics and control of ACMV and EACMCMV co-infections.MethodsA deterministic compartmental model describing cassava plants and whitefly vectors was formulated to capture single infections and co-infection dynamics. Control measures, including resistant varieties, disease-free planting materials, modern agricultural practices, and vector removal, were incorporated. Equilibrium and stability analyses were performed, and basic reproduction numbers were derived using the next-generation matrix method. Numerical simulations were conducted to support the analytical results.ResultsThe disease-free equilibrium is locally and globally asymptotically stable when the relevant basic reproduction number is less than one, indicating possible eradication under effective control. When it exceeds unity, endemic and oscillatory dynamics may occur. The reproduction number for the co-infection model exceeds those of the single-virus models, suggesting greater persistence and severity. Simulations show that combining resistant varieties and vector control substantially reduces infection levels.DiscussionIntegrated control strategies are essential to reduce co-infection burdens and maintain reproduction numbers below unity. The model provides quantitative guidance for improving cassava disease management and supporting food security.
BackgroundPublic health policy and disease surveillance systems require accurate forecasting of infectious disease dynamics to support timely interventions and resource allocation. However, classical linear time-series models often fail to capture abrupt regime shifts, nonlinear transmission patterns, and heterogeneous reporting commonly observed in surveillance data.MethodsThis study investigates the practical advantages of hierarchical Bayesian smooth transition autoregressive (BH-STAR) models, including logistic (LSTAR) and exponential (ESTAR) specifications. Performance is evaluated through extensive simulation studies under controlled nonlinear data-generating mechanisms, followed by an empirical application to COVID-19 surveillance data from 53 African countries collected between March 2020 and December 2022.ResultsSimulation studies revealed a key directional asymmetry in model misspecification: fitting a logistic transition function to data generated under exponential dynamics resulted in moderate, stable parameter bias, whereas fitting an exponential transition function to logistic dynamics induced severe, compounding bias. Despite this parameter confounding under model misspecification, predictive accuracy remained stable across both data-generating processes. In the empirical application, the BH-STAR models consistently outperformed linear alternatives in out-of-sample forecasting. The hierarchical logistic STAR (HLSTAR) model achieved the highest overall predictive accuracy, reducing validation errors to an MAE of 0.58 and a MdAPE of 12.8%, corresponding to country-level forecast error reductions of 30–50% and overall error reductions exceeding 50% relative to the standard autoregressive benchmark.ConclusionHierarchical Bayesian smooth transition autoregressive models provide accurate forecasts by accommodating nonlinear regime-switching dynamics while delivering robust uncertainty quantification. These features make them well suited for infectious disease surveillance and public health decision-making in resource-limited, high-uncertainty settings.
IntroductionAntibody-dependent enhancement (ADE) creates complex multi-pathogen dynamics that fundamentally alter extinction probabilities in co-circulating dengue and Zika viruses.MethodsWe develop a stochastic framework for five co-circulating pathogens (dengue serotypes 1 -4 and Zika) with asymmetric ADE. Using WKB large-deviation theory, we compute extinction actions and most-probable pathways via Minimum Action, Nudged Elastic Band, and adaptive SVD methods. WKB actions are validated against Monte Carlo simulations (Pearson r= 0.98, max relative error 0.3%).ResultsThe model predicts that DENV4 acts as a keystone pathogen: its elimination triggers a cascade (DENV4 → DENV2 → Zika) that is exponentially more probable than direct multi-pathogen extinction. The parameter space separates into three extinction regimes –sequential, pairwise, and total collapse –with hysteresis near η≈2.8 Sensitivity analysis shows keystone identity is stable under ±20% ADE matrix perturbations. In a symmetric n-pathogen limit on the symmetric invariant manifold, single-serotype extinction action decreases as 1/n while global elimination action grows as n·𝒮sym.DiscussionWithin this symmetric idealization, this provides a possible mechanistic reference point for understanding how serotype diversity could simultaneously promote local vulnerability and global persistence. Whether a similar relationship holds in the fully asymmetric dengue-Zika system is an open question. Limitations include the direct-transmission approximation (quantitative error < 15%) and time-averaged ADE. All predictions are model-based and require empirical validation.
HIV persistence within germinal centers is driven by complex interactions among T follicular helper (Tfh), T follicular regulatory (Tfr), and follicular CD8+ (fCD8+) T cells. Yet, the mechanisms sustaining the viral reservoir remain poorly understood. We developed a mechanistic within-host mathematical model to quantify the coupled regulations of Tfh–Tfr–fCD8+ dynamics governing viral replication, immune control, and reservoir stability. Simulations revealed that Tfh cells provide structural support for sustained viral production and that disrupting their function reduces viral load and may collapse the reservoir. Tfr cells primarily modulated the amplitude and timing of infection, buffering Tfh-driven viral expansion, whereas cytotoxic CD8+ T cells reduced the infection burden but could not independently achieve eradication. The nonlinear nature of germinal center immune control was demonstrated by findings that increasing infected-cell lysis increased viral load and that deregulation of regulatory mechanisms altered infection dynamics in a different cell line. To destabilize the reservoir and guide sensible HIV cure therapies, effective therapeutic approaches must focus on network structure and helper-cell support rather than isolated immune components. We can better understand how the immune network architecture controls HIV replication, immunological containment, and long-term reservoir structure by applying mechanistic immunology and quantitative analysis. This theoretical framework can guide the strategic development of therapeutic approaches.
We investigate a coupled eco-epidemiological model that integrates predator–prey dynamics with a Susceptible–Infected–Susceptible (SIS) disease framework. To examine the effects of parameter variability, we perform large-scale numerical simulations under three regimes: variation in epidemiological parameters, variation in ecological parameters, and simultaneous variation in both. For each case, we analyze equilibrium population levels of susceptible and infected prey and predators, along with convergence time. The results reveal distinct system behaviors across the three regimes. Variation in epidemiological parameters leads to slower convergence and occasional long transients, while ecological variation results in faster stabilization with generally low infection levels. When all parameters vary simultaneously, the system exhibits the widest range of outcomes, including both rapid convergence and rare prolonged dynamics. Across all cases, recovery rates play a dominant role in regulating infection levels, whereas convergence time is shaped by the combined influence of multiple parameters. To further interpret these patterns, we apply statistical methods and factor analysis to identify the low-dimensional structure underlying the system dynamics. These findings provide insight into how interactions between ecological and epidemiological processes shape long-term population outcomes under parameter uncertainty.
IntroductionObservational data collected from electronic health records (EHR) are often obtained at irregular visit time points because the visit process may depend on the patient's medical condition or disease severity, which is usually the outcome of interest. The use of traditional models, such as the linear mixed model, to analyse such outcomes has been shown to produce biased estimates. In this study, we aim to determine the extent to which irregular visits warrant the use of alternative models for analyzing extreme irregular longitudinal data.MethodsWe compared the performance of the linear mixed model (LMM), broken stick model (BSM), generalized estimating equation (GEE), and weighted GEE based on bias, coverage probability, standard error, and mean squared error using simulated data. The simulated data were generated under extreme irregular visit, and extreme irregular visit with missingness scenarios, with a moderate sample size (n = 500). The performance of these models was also assessed using real data from a study of patients who underwent metabolic and bariatric surgery at Tygerberg Hospital.ResultsUnder extreme irregular visits, the LMM showed varying performance depending on the degree of informativeness. When the degree of informativeness was set to 0.5, the LMM achieved the smallest ARB (1.3%) and an acceptable coverage probability (94%) compared with the other models. Under extreme irregular visit patterns with missingness, all models performed well at 10% and 20% missingness proportions; however, BSM and weighted GEE did not perform well at 40% missingness. For the real data set, the models produced similar estimates of the treatment effect. This may be because the real data exhibited moderate to no irregularity, as suggested by the Pearson correlation between the gap times and the longitudinal outcome.ConclusionThe study findings revealed varying performance of LMM across degrees of informativeness, with the smallest ARB and an acceptable coverage probability achieved when the degree of informativeness is set to 0.5 for extreme irregular visits only. Finally, under extreme irregular visits with missingness, the LMM, weighted GEE, and GEE are preferred at missingness proportions of 10% and 20%, whereas GEE and LMM are preferred at 40%.
BackgroundThe Global Youth Tobacco Survey is one of the most important sources of data on adolescent tobacco use worldwide. However, studies using these data often apply inconsistent statistical methods, particularly in how they handle complex survey designs and select predictors for analysis. Many analyses use simplified approaches or proprietary software, making results difficult to reproduce or compare across countries and survey years. We developed a clear, open-source workflow to guide researchers in selecting predictors and modeling adolescent smoking outcomes in a way that is transparent, theory-informed, and reproducible in both R and Python.ResultsThe framework is designed to incorporate the full two-stage survey design, including weights, stratification, and clustering, with the R implementation serving as the reference platform for design-based inference. Key demographic factors (age, sex, grade, and region or country) are retained in all models, while other modifiable predictors are selected using a constrained stepwise procedure guided by model fit. We demonstrate the approach using Zambia 2021 and pooled data from Ghana, Mauritius, Seychelles, and Togo, 2015–2019. The pooled dataset included 15,914 adolescents, of whom 13,360 had complete information for the complete-case modeling analysis. When identical model specifications were applied, the R and Python implementations selected the same final model and produced nearly identical adjusted odds ratios, with differences below 0.01. However, standard errors and confidence intervals differed slightly because Python's implementation relied on survey weights only and did not fully account for clustering and stratification. In the current cigarette-smoking model, the final model showed adequate event support, with 1,479 current smokers and 33 estimated parameters, giving an events-per-variable value of 44.82. Factors independently associated with current cigarette smoking included intention to use tobacco in the next 5 years, ever trying cigarette smoking, ownership of an item with a tobacco logo, seeing teachers smoking at school, and being taught about the dangers of tobacco use.ConclusionsThis study provides a practical and generalizable framework for analysing adolescent smoking using complex survey data. By combining theoretical grounding, transparent variable selection, and open-source tools, the workflow improves reproducibility, cross-country comparability, and policy relevance. It can be readily adapted to other large-scale health surveys to strengthen evidence for tobacco control and prevention efforts.
In this work, we developed an Interdependent Emission State's Hidden Markov model (IES-HMM) for studying the spread of COVID-19 disease among southern states, namely Andhra Pradesh, Karnataka, Kerala, Telangana, Tamil Nadu, and Pondicherry. The model incorporates two distinct components: invisible states and emission states. COVID cases registered in southern states other than Tamilnadu will be classified as “invisible,” and those reported in Tamilnadu will be emission states. In this IES-HMM, we assume that emission states are influenced by invisible and emission states. This study also aims to explore probability distributions for 3-day sequences based on the conditional probabilities between invisible and emission states. The derivation of all mathematical relations for measuring the different statistical characteristics of the developed model is another significant contribution of this study. The developed model will give the exact result. This study can also be used to formulate optimization models for effective healthcare management during the treatment of the disease.
IntroductionFood-borne diseases remain a major public health challenge, particularly in low- and middle-income countries where inadequate sanitation, poor livestock management, and limited healthcare access facilitate disease persistence. Among these diseases, Taenia solium cysticercosis is a neglected tropical disease associated with substantial morbidity, economic losses, and social disruption. This study develops a multiscale mathematical framework to investigate the transmission dynamics of T. solium across human and pig populations.MethodsA coupled multiscale model was formulated by integrating within-host parasite dynamics in humans and pigs with between-host population-level transmission processes. The model captures the complete lifecycle of T. solium and was analyzed using mathematical and epidemiological techniques. The basic reproduction number, R0 was derived, sensitivity analyses were performed to identify influential parameters, and numerical simulations were conducted using non-standard finite difference schemes.ResultsThe analysis showed that R0 depends on parameters from both within-host and between-host scales, demonstrating a strong reciprocal relationship between parasite development and population-level transmission. Sensitivity analysis identified key within-host developmental rates and between-host transmission parameters as the most influential drivers of disease persistence. Numerical simulations further revealed that alterations in within-host processes significantly affect population-level infection dynamics, while changes in transmission parameters directly influence within-host parasite burden.DiscussionThe findings highlight the importance of integrated control strategies that simultaneously target infected individuals, livestock management practices, and environmental contamination. By explicitly linking within-host and between-host processes, the proposed multiscale model provides an evidence-based framework for understanding T. solium transmission and supports the design of effective taeniasis and cysticercosis control programmes in endemic regions. The model offers valuable insights for policymakers and public health practitioners seeking sustainable disease management strategies.
This paper investigates the one-dimensional bin packing problem with time windows (1DBPP-TW), a practical variant of the bin packing problem widely applied in logistics. Existing relevant studies mainly focus on bin packing problems with variable-sized bins under time constraints, while 1DBPP-TW with homogeneous bins and a shared time window requirement for items in one bin has not been fully explored. Given the NP-completeness of the bin packing problem, exact algorithms are not applicable to large-scale instances, so this paper first establishes a mathematical model for 1DBPP-TW and generates two benchmark datasets including optimal-solution-known instances and random instances for verification, then proposes two efficient algorithms: the Greedy on Time Range (GTR) heuristic which ensures real-time response and obtains reliable solutions within 0.001 seconds for large instances, and the Iterative Local Search (ILS) metaheuristic equipped with three neighborhood operators to improve convergence rate and reduce the number of used bins by an average of 7.3% compared with GTR. Comparative experiments with the CPLEX solver within a 3600-second time limit show that the two proposed algorithms have higher computational efficiency and equivalent or better solution quality for medium and large instances. In practice, these algorithms resolve logistics industry problems including low loading efficiency and time window conflicts, and theoretically, this research advances the combinatorial optimization theory of time-constrained bin packing variants and provides standard benchmarks for follow-up studies. This study is limited to homogeneous bins and synthetic test instances, and future work will expand to multi-dimensional packing, dynamic time window constraints and practical verification with real industrial data.
Ordinal Likert-type indicators are pervasive in behavioral and social-science research, yet they violate the continuity and multivariate-normality assumptions that underlie many conventional factor-analytic procedures. Robust ordinal confirmatory factor analysis (CFA) estimators such as weighted least squares mean and variance adjusted (WLSMV) and diagonally weighted least squares (DWLS) reduce parameter bias, but they remain fundamentally linear and may leave threshold-related nonlinearities or item-level interactions in the residual structure. This study proposes a Hybrid Factor Analysis (HFA) framework that integrates robust CFA with machine-learning and deep-learning item signals through an adaptive weighting parameter, ω. The hybrid measurement weight, λ_combined, is constructed as a convex combination of normalized CFA loadings and normalized computational item-importance measures. The framework was evaluated in a Monte Carlo design for a five-factor, 36-indicator ordinal model under sample sizes of n = 250, 500, and 1,000, and then examined empirically using the Green Consumption Behavior dataset from Malaysia (n = 375). Under highly skewed thresholds and n = 250, HFA reduced loading mean squared error (MSE) from 0.092 to 0.068 relative to standalone WLSMV, a 26.1% improvement. In the empirical application, HFA improved global fit from CFI = 0.942, TLI = 0.938, and RMSEA = 0.068 under standalone WLSMV to CFI = 0.965, TLI = 0.959, and RMSEA = 0.048, while reducing the average bootstrap confidence-interval width from 0.124 to 0.089. The hybrid model also improved predictive performance over the linear CFA baseline (R2 = 0.32 vs. 0.28) while preserving confirmatory interpretability. These findings position HFA as a next-generation framework for ordinal psychometrics in which theory-constrained measurement and computational learning are treated as complementary rather than competing sources of information.
The phylophagous beetle (Agelastica alni) larvae outbreak defoliate the canopy in Black alder (Alnus glutinosa) dominated forests. Phytochemical defense of plants provides resistance against further beetle outbreaks. A tachinid parasitoid (Meigenia mutabilis) of the beetle, and optimal foraging of generalist passeriforms, provide resilience against the outbreak. Vernal growth enables recovery of the canopy. These resistance, resilience, and recovery processes operate at different time scales and are subjected to climate and management-dependent phenology. Under phenological variability, this eco-epidemic system of plant-pest-parasitoid-predator can adapt its stability or shift its regime from coexistence. In this study, we developed a process-based hybrid seasonal–annual model to examine the adaptive stability against regime shift of this system across nine phenological scenarios. We then applied optimal control theory to compare the cost and benefit of three management strategies against regime shift: parasitoid release, parasitoid release plus larval removal, and a fully integrated strategy including bird-habitat enhancement. Our results showed that, across the phenological variability, the coexistence equilibrium was the least frequently stable regime, whereas the parasitoid-free state was the most frequent. Only the fully integrated management delivered a locally stable managed equilibrium at minimum cost to maintain the coexistence equilibrium.
The existence of a weak solution of the incompressible isothermal Navier-Stokes equation with given initial conditions in the Sobolev space ℋ−2 for energy fluctuations and in ℋ−3 for enstrophy fluctuations is assumed. The existence, if it exists, is uniform with respect to the Euler limit of zero viscosity ν. Thus, the existence of weak solutions to the Euler equation with given initial conditions is established. These Euler solutions are the zero-viscosity limit of Navier-Stokes solutions, provided the Navier-Stokes solutions exist.
Accurate electricity consumption forecasting is essential for energy planning, particularly when the available data are limited, non-linear, and seasonally fluctuating. Although the classical Grey Model, GM(1,1), is effective for small-sample forecasting, its monotonic structure limits its ability to capture monthly seasonal variations. To address this limitation, this study develops a Modified Fractional Hausdorff Data Grouped grey Model, MFHDGGM(1,1), for forecasting Malaysia's monthly electricity consumption from 2011 to 2021. The novelty of the proposed model lies in integrating seasonal data grouping, fractional Hausdorff-order accumulation-based flexibility, and modified grey modeling within a unified forecasting framework. This structure improves the model's ability to capture both short-term seasonal variability and long-term consumption trends. The proposed model is compared with GM(1,1), DGGM(1,1), FHGM(1,1), FHDGGM(1,1), MGM(1,1), and MFHGM(1,1) using MAPE and RMSE. The empirical results show that MFHDGGM(1,1) achieves the best testing performance, with the lowest MAPE of 3.892% and RMSE of 667.349. In comparison, the testing MAPE values of GM(1,1), DGGM(1,1), FHGM(1,1), FHDGGM(1,1), MGM(1,1), and MFHGM(1,1) are 5.833%, 6.838%, 5.130%, 4.008%, 6.007%, and 5.057%, respectively. These findings confirm that the proposed modified fractional Hausdorff data grouped structure provides a more accurate and stable forecasting framework for seasonal electricity consumption under limited data conditions.
