The Heat Equation is a well-known partial differential equation that can be solved numerically by finite difference methods in time coupled with finite element methods in space. Crank-Nicolson methods can therefore be applied as finite difference methods and coupled with linear Lagrange finite element methods in order to solve the Heat equation and obtain an efficient convergence rate, the Heat equation is said to be solved by a Crank-Nicolson finite element method. However, the convergence rate of the Crank-Nicolson finite element method for the Heat equation can be affected if the exact solution entails time singularities; in that case the lack of smoothness of the solution though it is local in time, affects the convergence of the finite element method in the whole domain. This paper presents a Crank-Nicolson finite element method coupled to a Predictor-corrector algorithm to recover the optimal convergence rate when the solution has time singularities. The finite element method presented is based on the approximation of the time singular functions using a Fourier decomposition of the exact solution that leads to computable formulas of the time dependent coefficients of singularities that reduce the smoothness of the solution; so removing those coefficients ameliorate the efficiency of the Crank-Nicolson finite element method. Numerical experiments are presented to show the efficiency of the method.
The root-finding problem is one of the most important problems in Numerical Analyis. It arises in a wide variety of practical applications in physics, chemistry, biosciences, engineering, etc. As a matter of fact, determination of any unknown appearing implicitly led to the evolution of root-finding problem. Therefore, this paper focuses on the modification and analysis of a high order derivative-free iteration methods for finding roots of nonlinear algebraic equations of the form f (x) = 0. The methods require only one initial approximation. The proposed method is seen as an extension of the second-order Steffensen’s scheme, which is an Iterative method for approximating roots of non-linear equations which often breaks down when the derivative of the function value is zero or near zero at the point of iteration. This work therefore, seeks to introduce a method that overcomes such breakdown. The method herein is a combination of forward difference formula with Simpson’s quadrature in spirit of Steffensen. The idea is to modify the Steffensen’s method, which were recently developed to obtain derivative-free methods. The modified methods are shown to converge. We also describe how to obtain derivative-free methods to find solutions to multiple roots. Several numerical examples are provided to validate the theoretical order of convergence for nonlinear functions with simple roots and results obtained show the comparative advantage the proposed method has over well-known methods.
Abstract: A dominating set of a connected graph G = (V, E) is a subset D of V (G) such that every vertex of G is either in D or adjacent to a vertex in D . A resolving set of a connected graph G is a subset D of V (G) such that each vertex v of G has a unique representation with respect to D . A dominant resolving set of a graph G is a subset D of V (G) that resolves all vertices of G and dominates G . The dominant metric dimension, Ddim(G) , is the cardinality of the smallest such set. The dominant metric dimension of a graph combines metric dimension (location identification) with domination (coverage), making it ideal for sensor networks that require both unique positioning and full coverage. Similarly, a dominant edge resolving set of a connected graph G is a vertex subset D of V that resolves all edges and is a covering of G. Star fan graphs are composite graph structures formed by attaching fan graphs to the pendant vertices of a star graph, creating hierarchical networks that are useful for modeling hub-and-spoke systems with additional path structures. This paper computes the dominant metric dimension and dominant edge metric dimension of star fan graphs.
In the context of the ongoing digital transformation of governmental and corporate information systems, the development of intelligent document management solutions capable of efficient processing, structuring, and analysis of textual data has become increasingly important. Particular challenges arise in the processing of multilingual data and low-resource languages, such as Tajik, due to the limited availability of annotated corpora. The aim of this study is to develop and formalize a mathematical model of an intelligent document management system based on microservice architecture and transformer-based natural language processing techniques. The proposed approach integrates a distributed microservice architecture using gRPC with a named entity recognition (NER) model based on multilingual BERT. To address data scarcity, a synthetic data generation mechanism is introduced to augment the training corpus. The NER task is formulated as a probabilistic sequence labeling problem, and the training procedure includes fine-tuning of the transformer model and comparison with baseline approaches, including rule-based methods, Conditional Random Fields (CRF), and BiLSTM-CRF models. Experimental evaluation is conducted on a curated corpus of Tajik-language documents, divided into training, validation, and test subsets. The results demonstrate that the proposed model achieves an F1-score of 0.93, outperforming all baseline methods. In addition, the system exhibits near-linear scalability under horizontal scaling conditions and ensures fault tolerance through a hybrid mechanism that switches to a rule-based extractor in case of service unavailability. The proposed model provides a scalable and robust framework for intelligent document processing systems and can be effectively applied in governmental and corporate environments undergoing digital transformation.
Let ℋ be a finite commutative ring with unity. Let ℐ be a proper ideal of ℋ and 𝒮 is the multiplicative closed subset of ℋ which is disjoint with ℐ. The weakly S-prime ideal graph denoted by Gℐ(ℋ) is the undirected graph whose vertex set is the set of elements 𝔢 of ℋ such that the non-zero product ef is in ℐ and either se is in ℐ or sf is in ℐ for some f in ℋ and the two distinct vertices 𝔢 and 𝔣 are connected by an edge if and only if either se is in ℐ or sf is in ℐ for some s in 𝒮. The purpose of this article is to investigate the graph theoretic properties of the weakly S-prime ideal graph associated with ℋ. This study focuses on rings of order 2𝔭, 3𝔭 and 𝔭𝔮, where 𝔭 and 𝔮 are distinct primes. For these rings, the weakly S-prime ideal graph is a special type of graph and it is explained with examples. Furthermore, the graph theoretic concept of the weakly S-prime ideal graph Gℐ(ℋ) namely its girth, diameter, radius and size are studied. The relation between the weakly S-prime ideal graph and annihilator ideal graph associated with a ring of order 2𝔭 is described and it is proved that these two graphs are isomorphic.
Classical finite element methods (FEM) approximate curved boundaries by piecewise polynomials, which introduces geometric errors that degrade the accuracy of wave propagation simulations. Isogeometric analysis (IGA) overcomes this limitation by using the same B-spline basis functions for both exact geometry representation and solution approximation. This paper presents a complete Python implementation of IGA for solving the wave equation on curved domains, extending previous work from rectangular to curved geometries. The methodology includes three main steps. First, we construct exact B-spline parameterizations of a semicircular membrane and a quarter-annular plate, ensuring no geometric approximation error. Second, we discretize the weak Galerkin formulation in space using B-splines and in time using a fourth-order Runge-Kutta scheme. Third, we derive a generalized Courant-Friedrichs-Lewy (CFL) condition that incorporates the effective element size to account for the non-uniform Jacobian of curved mappings. Numerical experiments demonstrate that the method achieves optimal convergence rates. For quadratic B-splines, the observed rate reaches 2.98, very close to the theoretical value of 3. On the quarter-annulus test case, the IGA solution matches the analytical frequency with only 0.3% error, whereas standard FEM with a comparable number of degrees of freedom yields 2.1% error-a sevenfold improvement. The numerical CFL limit is accurately predicted by the generalized condition. The complete open-source Python code is provided in the appendix, enabling full reproducibility. This work lays a foundation for accurate wave propagation simulations on curved geometries in acoustics, elastodynamics, and seismology.
This work deals with two mathematical aspects of subsurface flow problems within aquifer systems, namely Mathematical Modeling and Theoretical Analysis. Concerning the Mathematical Modeling, the classical challenges for this class of problems are a rigorous description of diverse interactions that may take place between different involved aquifers. Recall that the main challenge at this stage is a realistic description of the flows from one aquifer to another passing necessarily through an aquitard which is a porous layer with small permeability coefficient and small thickness (compared with the mean thickness of involved aquifers. In the same way as most of flow phenomena, the governing equations of subsurface flows are based upon conservation laws.) To address the mathematical modeling of water exchange between different aquifers separated by aquitards we expose a mathematical approach based upon the Taylor expansion. Introducing the concept of observers located inside the aquitard and the neighboring aquifers, the mass conservation law has been applied and has led to one mass balance equation for each aquifer. Thanks to this original approach we have recovered the well-known mass balance equations exposed in the literature for flow problems in aquifer systems. Due to the assumptions of small thickness and homogeneity of the absolute permeability of aquitards for our framework the water flow is supposed vertical in aquitards and so we deal with one-dimensional flows there. This is the reason why the Taylor expansion deployed there concerns only the vertical space variable. The flux continuity has been applied to get the coupling of flow equations in the two aquifers. Since the flows in aquifers are supposed horizontal it is clear that the interface aquitard/aquifer flux acts as an additional source-term for each aquifer (and not a boundary term). Concerning the Theoretical Analysis of the global system of elliptic equations (as the flow is supposed to be submitted to a steady state) the Schauder Fixed Point Theorem has been applied for facing the nonlinearity of the right-hand sides of the system. This is the way we have got the existence of a solution to the system, but not the uniqueness. Thanks to a monotonicy assumption on the right-hand side vector-function we get the uniqueness of the solution. Finally the stability of that solution has been established under appropriate conditions.
This study focuses on the analysis of the flow behaviour of a chemically reacting Williamson fluid over an inclined rotating surface, taking into account the combined effects of Coriolis force and Joule heating. A mathematical model is developed to describe the fluid dynamics by integrating the equations of momentum, energy, and species concentration under the influence of these physical phenomena. The governing partial differential equations have been non-dimensionalised and transformed into a system of ordinary differential equations by introducing similarity transform. The resulting ODEs are written as a truncated series whose coefficients are obtained by using the collocation numerical technique on the resulting equations. The flow variables are determined and presented in profile and tabular form. The results were validated by comparing with MATLAB bvp4c and the error ranges between 0.09% and 0.61%, indicating that the method of solution is admissible. It is established that the Coriolis force enhances fluid velocity while simultaneously reducing both temperature and concentration. In contrast, Joule heating increases temperature but decreases fluid velocity and concentration. Additionally, chemical reactions lead to a reduction in velocity and concentration due to the consumption of reactive species, while simultaneously increasing temperature due to exothermic effects.
This study examines the coupled water-sediment energy dynamics in dam-driven closed-conduit flows, aiming to forecast discharge capacity and optimize release strategies for water supply, flood control, and hydropower generation. Sediment accumulation within dams significantly reduces hydraulic efficiency, limits the water supply, and undermines the overall benefits of dam operations by causing tunnel blockages, energy losses, and low discharge conditions. The novelty of this work is to develop a mathematical model that incorporates water-sediment interactions to accurately predict deposition zones, quantify energy dissipation, and support effective sediment management strategies. The governing equations are the continuity equation, the momentum equation, the energy equation, and the concentration equation. These equations are transformed from nonlinear partial differential equations into a system of linear ordinary differential equations using similarity transformations. The resulting equations are then solved using the collocation numerical technique and simulated in MATLAB software to obtain the profiles of the flow variables. The flow variables profiles are presented graphically. Flow parameters are varied, and their effects on the flow variables are discussed. It was observed that an increase in both the Reynolds number and the thermal Grashof number leads to an increase in velocity profiles, whereas an increase mass Grashof number produces an opposite effect by reducing the fluid velocity. Temperature of the fluid decreases with increasing Prandtl number, while an increase in the Eckert number leads to higher temperature profiles. The concentration profile decreases as the Schmidt number, Concentration ratio, and thermophoresis parameter increase. The research findings can help in making informed decisions on the in-dam safety, improving sediment management practices, ensuring reliable hydropower generation, and preventing blockage of the pipe. Furthermore, the research contributes to the design of more resilient discharge structures that can efficiently handle sediment, thereby extending the lifespan of hydraulic infrastructure and promoting sustainable operation of dam-driven systems.
. In this paper, we study different types of ultimate dynamics of the three-dimensional breast cancer model that describes the interactions between tumor cells, cytotoxic T cells, and helper T cell populations. We explore inner equilibrium points and two subsystems: The first one is a subsystem without cancer cells, the other is a subsystem without cytotoxic T cells. Using the localization theorem of compact invariant sets, we derive ultimate upper bounds for all cell populations and prove the existence of the attracting set. Employing these results, we derive several persistence conditions for all cell populations and cytotoxic T cell extinction conditions. Next, we localize compact invariant sets in the positive orthant with help of computing iteratively localization bounds. In particular, we find 1) conditions of the non-existence of inner compact invariant sets and 2) describe the iterative procedure for the narrowing of the sequence of localization polytopes. If the limit localization polytope is reduced to a point then this point is an asymptotically stable equilibrium point with the computed attraction domain. Our theoretical studies of inner ultimate dynamics are supplied by results of numerical simulation. Apart of this, the Hopf bifurcation case is investigated.
. This paper concerns a linearized stabilizer-free weak Galerkin finite element method (SFWGFEM) for the two-dimensional high-field (HF) model. The proposed method not only offers the flexibility to approximate functions and is compatible with meshes including hanging nodes as WG-FEM, but also maintains a simple format similar to the traditional finite element method. Optimal error estimations are obtained in both L2 and energy norms through the introduction of a weak Galerkin Ritz projection operator, where the associated Ritz projection error is derived by a dual problem. The superconvergent results between the Ritz projection of the exact function and the numerical solution in the energy norm lead to optimal L2 estimations. Numerical experiments are conducted to demonstrate the theoretical analysis.
. In this work, a method for transforming the double layer potential into a volume potential is presented. This new approach is applied to the study of non-classical boundary value problems for the Laplace equation, particularly those of Bitsadze-Samarskii type, where boundary conditions involve values inside the domain. The method also addresses boundary value problems with non-local boundary conditions, where traditional techniques are less effective. The results contribute to the broader theory of correct boundary extensions and restrictions of differential operators and offer a new tool for analyzing various elliptic problems.
. This paper develops innovative algorithms designed to address the complexity and uncertainty inherent in multiple attribute decision making (MADM). By employing intervalvalued intuitionistic fuzzy (IVIF) sets, the proposed approach offers structured solution procedures that enhance decision accuracy and reliability. Attribute information is expressed through IVIF numbers (IVIFNs), enabling decision makers to capture subtle nuances and ambiguities in the decision process. A central novelty is the introduction of a generalized score function (GSF) that overcomes the shortcomings of existing functions by incorporating a regular parameter "n"; (>= 1). The algorithms are founded on demand and evaluation functions: the demand function allows qualitative expression of preferences and requirements through logical connectives and attribute weights, while the evaluation function, derived via max-min operators, yields an IVIFN representation of each alternative. Two variants of the algorithm are presented-one using crisp attribute weights and the other using IVIFNs for greater flexibility. The evaluation function is expressed as an IVIFN, whose score values are used to rank alternatives and identify the optimal choice. Several illustrative examples validate the practicality and robustness of the approach. A real-world case study on sustainable crypto currency selection highlights its effectiveness in identifying the best alternative.
. This study proposes a novel hybrid multi-criteria decision-making approach based on Interval-Valued Fermatean Fuzzy sets (IVFFS) to evaluate information security risks in the SDN environments. The proposed model consists of four stages. First, the SDN architecture and its related vulnerabilities are identified. Second, cause-effect relationships among criteria are analyzed using IVFF-DEMATEL, and criterion weights are determined using IVFF-AHP. Third, the Software-Defined Networks (SDN) security risks are ranked through IVFF-TOPSIS. Finally, the stability and reliability of the proposed framework are validated through sensitivity and comparative analyses using IVFF-ARAS. The study evaluates five key security criteria (confidentiality, consistency, integrity, authentication, and availability) and 12 SDN-specific vulnerabilities. Results indicate that confidentiality is the most critical risk factor, whereas availability is the least prioritized. Sensitivity analysis confirms robust rankings under varying weights, and comparative results show high consistency with ARAS. Overall, the proposed approach provides a reliable decision-support tool for the SDN risk assessment under uncertainty.
Automated ischemic stroke segmentation remains difficult because non-contrast Computed Tomography (CT) is low contrast and noisy, whereas Diffusion Weighted Imaging (DWI) shows heterogeneous lesions. Conventional U-Net models rely on local receptive fields and unselective skip fusion, limiting global context and noise control. We propose GBA-Net, a UNet variant that combines a high-capacity gated Convolutional Neural Network (CNN) bottleneck for long range dependencies with convolutional block attention modules that refine multiscale features before decoder fusion. The bottleneck helps interpret subtle CT hypo densities and link scattered infarcts in DWI, while attention suppresses CT noise and filters high intensity mimics, improving boundary delineation. We evaluated GBA-Net on ISLES 2024 and TEKNOFEST 2021 and compared it with nine baselines including UNet, UNet++, DeepLabV3+, and Seg-Former. GBA-Net achieved Dice 0.7376 and 0.7140 and the best average ASSD of 4.73 pixels on CT.
We study some necessary and sufficient conditions for the boundedness of the fractional integral operator I alpha and its commutator [b, I alpha] on the total Morrey spaces Lp,lambda,& micro;(G), where G is a stratified Lie group. We characterize the strong and weak Spanne type and Adams type boundedness of I alpha on Lp,lambda,& micro;(G), respectively. We also give necessary and sufficient conditions for the boundedness of the commutator of the fractional integral operator [b, I alpha] on Lp,lambda,& micro;(G) when b belongs to the spaces BMO(G).
In this paper, a reaction-diffusion HIV coinfection model between CD4(+) T cells and macrophages with humoral immunity and recovery of infected CD4(+) T cell is proposed in heterogeneous environment. First, the existence and ultimate boundedness of global solutions are discussed, the basic reproduction number R-0 is obtained. Second, criteria on the globally asymptotic stability of infection-free steady state is established if R-0 < 1, the uniform persistence of disease is obtained if R-0 > 1 in the absence of humoral immune response. Furthermore, the immune response reproduction number R-1 is calculated under the homogeneous environment, by which criteria on the globally asymptotic stability of humoral immune-free equilibrium and chronic-infection equilibrium are obtained, respectively. Finally, the theoretical results are demonstrated by numerical simulations, we find that ignoring the HIV infection with macrophages will underestimate the viral loads of HIV within body, meanwhile, diffusion could cause great impact on HIV spread, it could change the situation of HIV infection at some positions from no infection to infection.
Problem considered: Reliable estimation of childhood malnutrition remains a major public health challenge in low-and middle-income countries, where large-scale surveys such as the Demographic and Health Surveys (DHS) often suffer from measurement error and data heterogeneity. Ignoring these issues can bias prevalence estimates and distort the identification of socioeconomic determinants. Methods: This study develops a hierarchical Bayesian logistic regression model that accounts for both measurement error and clustering effects by region and survey year. The model incorporates known sensitivity and specificity to adjust for outcome misclassification and includes random effects to capture between-region and temporal variability. Using simulated DHS-like data, the corrected model was compared to an uncorrected counterpart in terms of key performance metrics-prevalence, area under the Receiver Operating Characteristic(ROC) curve (AUC), and accuracy-across survey years (2004, 2011, 2018, and 2022). Results: The Bayesian correction improved predictive accuracy and reduced bias in prevalence estimates. The corrected model achieved consistently higher AUC values (0.882-0.930) compared to the uncorrected model (0.878-0.928), and exhibited lower mean squared error (0.121 vs. 0.137). The inclusion of regional and temporal random effects effectively captured unobserved heterogeneity. Posterior parameter estimates revealed several significant socioeconomic predictors influencing child malnutrition. Conclusion: The proposed Bayesian hierarchical framework demonstrates improved accuracy and robustness in estimating malnutrition prevalence when accounting for measurement error. These findings highlight the importance of error correction and multilevel modeling for more reliable health policy decision-making based on survey data.
In the complex domain of real-world farm planning, agricultural decision-makers are continually confronted with a myriad of interconnected challenges, particularly in water management, crop selection, the determination of optimal crop combinations, and the effective implementation of advanced agricultural techniques to boost production and ensure sustainability. These operational hurdles are not faced in isolation; they are deeply intertwined with broader concerns of socio-economic development, such as farmer livelihoods and regional food security, and are often intensified by significant resource scarcity, especially in arid and semi-arid regions. To address these critical issues, the robust mathematical framework of linear programming (LP) offers a powerful solution, providing a systematic methodology to optimize farm returns by ensuring the most efficient allocation of available, limited resources. The primary objective of this study is to develop a linear programming model tailored to the specific agricultural context of Tiruchirappalli District, Tamil Nadu. This model is designed to identify the optimal crop combination from a set of feasible alternatives and to determine precisely how critical resources—such as land and water—can be allocated to enhance overall productivity for the farming community. The comprehensive analysis was carried out using Linear Optimization Techniques, and the resultant model was formulated and solved using the specialized LINGO software to derive a practical, actionable, and data-driven solution for the district's agricultural stakeholders. The analysis reveals that the optimal resource allocation strategy can yield a maximum achievable productivity of 786,151,300 kg for the major crops in the district.
The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs and , their semi-strong product is a graph whose vertex set is , and the edge set is defined as follows: if and are two vertices in the semi-strong product , then there is an edge between them if and only if and and are adjacent in , or and are adjacent in and and are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.
