AppliedMath, Volume 5, Issue 3 (September 2025) – 23 articles

The objective of this investigation is to obtain numerical solutions for a variety of mathematical models in a wide range of disciplines, such as chemical kinetics, neurosciences, nonlinear optics, metallurgical separation/alloying processes, and asset dynamics in mathematical finance. This research features numerical simulations conducted with a remarkably low error measure, providing a visual representation of the examined models in these areas. The proposed method is the double Laplace–Adomian decomposition method, which facilitates the numerical acquisition and analysis of solutions. This paper presents the first report of numerical simulations employing this innovative methodology to address these problems. The findings are expected to benefit the natural sciences, mathematical modeling, and their practical applications, representing the innovative aspect of this article. Additionally, this method can analyze many classes of partial differential equations, whether linear or nonlinear, without the need for linearization or discretization. Full article

This paper develops a unified framework for mathematical finance under general semimartingale models that allow for dividend payments, negative asset prices, and unbounded jumps. We present a rigorous approach to the mathematical modeling of financial markets with dividend-paying assets by defining appropriate concepts of numéraires, discounted processes, and self-financing trading strategies. While most of the mathematical results are not new, this unified framework has been missing in the literature. We carefully examine the transition between nominal and discounted price processes and define appropriate notions of admissible strategies that work naturally in both settings. By establishing the equivalence between these models and providing clear conditions for their applicability, we create a mathematical foundation that encompasses a wide range of realistic market scenarios and can serve as a basis for future work on mathematical finance and derivative pricing. We demonstrate the practical relevance of our framework through a comprehensive application to dividend-paying equity markets where the framework naturally handles discrete dividend payments. This application shows that our theoretical framework is not merely abstract but provides the rigorous foundation for pricing derivatives in real-world markets where classical assumptions need extension. Full article

The Korteweg–de Vries (KdV) equation is a nonlinear evolution equation that reflects a wide variety of dispersive wave occurrences with limited amplitude. It has also been used to describe a range of major physical phenomena, such as shallow water waves that interact weakly and nonlinearly, acoustic waves on a crystal lattice, lengthy internal waves in density-graded oceans, and ion acoustic waves in plasma. The KdV equation is one of the most well-known soliton models, and it provides a good platform for further research into other equations. The KdV equation has several forms. The aim of this study is to introduce and investigate a (2+1)-dimensional generalized fifth-order KdV equation with power law nonlinearity (gFKdVp). The research methodology employed is the Lie group analysis. Using the point symmetries of the gFKdVp equation, we transform this equation into several nonlinear ordinary differential equations (ODEs), which we solve by employing different strategies that include Kudryashov’s method, the $(G^{\prime }/G)$ expansion method, and the power series expansion method. To demonstrate the physical behavior of the equation, 3D, density, and 2D graphs of the obtained solutions are presented. Finally, utilizing the multiplier technique and Ibragimov’s method, we derive conserved vectors of the gFKdVp equation. These include the conservation of energy and momentum. Thus, the major conclusion of the study is that analytic solutions and conservation laws of the gFKdVp equation are determined. Full article

This paper addresses the numerical evaluation of highly oscillatory integrals by developing a spectral Galerkin–Levin approach that efficiently solves Levin’s differential equation formulation for such integrals. The method employs Legendre polynomials as basis functions to approximate the solution, leveraging their orthogonality and favorable numerical properties. A key finding is that the Galerkin–Levin formulation is invariant with respect to the choice of polynomial basis—be it monomials or classical orthogonal polynomials—although the use of Legendre polynomials leads to a more straightforward derivation of practical quadrature rules. Building on this, this paper derives a simple and efficient numerical quadrature for both scalar and matrix-valued highly oscillatory integrals. The proposed approach is computationally stable and well-conditioned, overcoming the limitations of collocation-based methods. Several numerical examples validate the method’s high accuracy, stability, and computational efficiency. Full article

Hydropressor systems are of paramount importance in keeping water supplies running properly. A typical such device consists of two (or more) identical electropumps operating alternately, so as to avoid downtime as much as possible. We consider a dual pump configuration to identify the ideal usage proportion of each pump (from 0%-100%, meaning interchange only upon failure, to 50%-50%, where each pump works half the time) in order to improve availability, accounting solely for corrective maintenance. We also address the possibility of improving the availability of a single pump under the hazard of failure in three different ways (with their own occurrence frequencies), while also accounting for preventive maintenance. Both settings are tackled through Monte Carlo simulation and the models are implemented with the Python 3.12 programming language. The results indicate that significant improvements to standard industry practices can be made. Full article

We present a formal framework for modeling neural network dynamics using Category Theory, specifically through Markov categories. In this setting, neural states are represented as objects and state transitions as Markov kernels, i.e., morphisms in the category. This categorical perspective offers an algebraic alternative to traditional approaches based on stochastic differential equations, enabling a rigorous and structured approach to studying neural dynamics as a stochastic process with topological insights. By abstracting neural states as submeasurable spaces and transitions as kernels, our framework bridges biological complexity with formal mathematical structure, providing a foundation for analyzing emergent behavior. As part of this approach, we incorporate concepts from Interacting Particle Systems and employ mean-field approximations to construct Markov kernels, which are then used to simulate neural dynamics via the Ising model. Our simulations reveal a shift from unimodal to multimodal transition distributions near critical temperatures, reinforcing the connection between emergent behavior and abrupt changes in system dynamics. Full article

A novel algorithm was proposed for solving the max-cut problem, which seeks to identify the cut with the maximum weight in a given graph. Our technique is based on the bundle approach, applied to a newly formulated semidefinite relaxation. This research establishes the theoretical convergence of our approximation technique and presents the numerical results obtained on several large-scale graphs from the BiqMac library, specifically with 100, 250, and 500 nodes. The resulting performance was compared with that produced by two alternative semidefinite programming-based approximation methods, namely the BiqMac and BiqBin solvers, by comparing the CPU time and the number of function calls. The primary objective of this work was to enhance the scalability and computational efficiency in solving the max-cut problem, particularly for large-scale graph instances. Despite the development of numerous approximation algorithms, a persistent challenge lies in effectively handling problems with a large number of constraints. Our algorithm addresses this by integrating a novel semidefinite relaxation with a bundle-based optimization framework, achieving faster convergence and fewer function calls. These advancements mark a meaningful step forward in the efficient resolution of NP-hard combinatorial optimization problems. Full article

The process of feature selection is a critical component of any decision-making system incorporating machine or deep learning models applied to multidimensional data. Feature selection on input data can be performed using a variety of techniques, such as correlation-based methods, wrapper-based methods, or embedded methods. However, many conventionally used approaches do not support backwards interpretability of the selected features, making their application in real-world scenarios impractical and difficult to implement. This work addresses that limitation by proposing a novel correlation-based strategy for feature selection in regression tasks, based on a three-dimensional visualization of correlation analysis results—referred to as three-dimensional correlation graphs. The main objective of this study is the design, implementation, and experimental evaluation of this graphical model through a case study using a multidimensional dataset with 28 attributes. The experiments assess the clarity of the visualizations and their impact on regression model performance, demonstrating that the approach reduces dimensionality while maintaining or improving predictive accuracy, enhances interpretability by uncovering hidden relationships, and achieves better or comparable results to conventional feature selection methods. Full article

This study investigates the relationship between sovereign credit rating transitions and domestic equity market performance, focusing on Greece from 2004 to 2024. Although credit ratings are central to sovereign risk assessment, their immediate influence on financial markets remains contested. This research adopts a multi-method analytical framework combining algebraic combinatorics and time-series econometrics. The methodology incorporates the construction of a directed credit rating transition graph, the partially ordered set representation of rating hierarchies, rolling-window correlation analysis, Granger causality testing, event study evaluation, and the formulation of a reward matrix with optimal rating path optimization. Empirical results indicate that credit rating announcements in Greece exert only modest short-term effects on the Athens Stock Exchange General Index, implying that markets often anticipate these changes. In contrast, sequential downgrade trajectories elicit more pronounced and persistent market responses. The reward matrix and path optimization approach reveal structured investor behavior that is sensitive to the cumulative pattern of rating changes. These findings offer a more nuanced interpretation of how sovereign credit risk is processed and priced in transparent and fiscally disciplined environments. By bridging network-based algebraic structures and economic data science, the study contributes a novel methodology for understanding systemic financial signals within sovereign credit systems. Full article

The development of solar technologies and the widespread adoption of photovoltaic (PV) panels have significantly transformed the global energy landscape. PV panels have evolved from niche applications to become a primary source of electricity generation, driven by their environmental benefits and declining costs. However, the performance and operational lifespan of PV systems are often compromised by various faults, which can lead to efficiency losses and increased maintenance costs. Consequently, effective and timely fault detection methods have become a critical focus of current research in the field. This work proposes an innovative video-based method for the dimensional evaluation and detection of malfunctions in solar panels, utilizing processing techniques applied to aerial images captured by unmanned aerial vehicles (drones). The method is based on a novel mini-pattern matching algorithm designed to identify specific defect features despite challenging environmental conditions such as strong gradients of non-uniform lighting, partial shading effects, or the presence of accidental deposits that obscure panel surfaces. The proposed approach aims to enhance the accuracy and reliability of fault detection, enabling more efficient monitoring and maintenance of PV installations. Full article

Pedestrian flow simulations play a pivotal role in urban planning, transportation engineering, and disaster response by enabling the detailed analysis of crowd dynamics and walking behavior. While physical models such as the Social Force model and Boids have been widely used, they often struggle to replicate complex inter-agent interactions. On the other hand, reinforcement learning (RL) methods, although adaptive, suffer from limited interpretability due to their opaque policy structures. To address these limitations, this study proposes a pedestrian simulation framework based on the Anticipatory Classifier System 2 (ACS2), a rule-based evolutionary learning model capable of extracting explicit behavior rules through trial-and-error learning. The proposed model captures the interactions between agents and environmental features while preserving the interpretability of the acquired strategies. Simulation experiments demonstrate that the ACS2-based agents reproduce realistic pedestrian dynamics and achieve comparable adaptability to conventional reinforcement learning approaches such as tabular Q-learning. Moreover, the extracted behavior rules enable systematic analysis of movement patterns, including the effects of obstacles and crowd composition on flow efficiency and group alignment. The results suggest that the ACS2 provides a promising approach to constructing interpretable multi-agent simulations for real-world pedestrian environments. Full article

This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values. Full article

Robust mean-geometric mean (MGM) linking methods enable reliable group comparisons in item response theory models under fixed and sparse differential item functioning. This article evaluates six alternative standard error and confidence interval (CI) estimation methods across four MGM linking approaches. Our Simulation Study demonstrates that CIs based on the delta method or bootstrap procedures using the normal distribution or empirical quantiles exhibit highly inflated coverage rates. In contrast, CIs derived from a weighted least squares estimation problem, as well as basic and bias-corrected bootstrap methods, yield satisfactory coverage rates in most simulation conditions for robust MGM linking. Full article

This work explores the morphogenesis of the skeletal mineral component, with a specific emphasis on hydroxyapatite (HAp) crystal assembly. Bone is fundamentally a triphasic biomaterial, consisting of an inorganic mineral phase, an organic matrix, and an aqueous component. The inorganic phase (hydroxyapatite), is characterized by its hexagonal prismatic nanocrystalline structure. We leverage a cellular automata (CA) paradigm to computationally simulate the mineralization process, leading to the formation of the bone’s hydroxyapatite framework. This model exclusively considers the physicochemical aspects of bone formation, intentionally excluding the biological interactions that govern in vivo skeletal development. To optimize computational efficiency, a simplified anatomical segment of a long bone (e.g., the femur) is modeled. This geometric simplification encompasses an outer ellipsoidal cylindrical boundary (periosteal envelope), an inner ellipsoidal surface defining the interface between cortical and cancellous bone, and a central circular cylindrical lumen representing the medullary cavity, which accommodates the bone marrow and primary vasculature. The CA methodology is applied to generate the internal bone microarchitecture, while deliberately omitting the design of smaller, secondary vascular channels. Full article

This research delves into the assessment of students’ perspectives regarding virtual internships within Brazilian organizations, a phenomenon accelerated by the global pandemic. Evaluating 78 students’ virtual internships via a survey, the study employs the Fuzzy TOPSIS Class method for analysis. Additionally, a sensitivity analysis was conducted to assess the robustness of the results. Key insights for enhancing virtual internships encompass: emphasizing application and deeper understanding of topics learned during the undergraduate course, enhancing understanding about how organizations work, and fostering comprehension of market dynamics. Among the points best rated by students are the opportunity to explore new subjects, development of soft skills, and supervisors’ competence in managing teams in virtual environments. This paper contributes methodologically by proposing a multicriteria decision-making approach to assess virtual internships. The findings serve as a valuable resource for internship supervisors in companies and higher education institutions, aiding them in guiding students through this pivotal developmental phase that shapes their future careers. Full article

Given a triangular mesh in ${\mathbb{R}}^3$ with a family of points associated with its vertices along with vectors associated with its edges, we propose a novel technique for the construction of locally generated fitting parametrized G1-spline interpolation surfaces. The method consists of a G1 correction over the mesh edges of the mesh triangles, produced using reduced side derivatives (RSDs) introduced earlier by the author in terms of the barycentric weight functions. In the case of polynomial RSD shape functions, we establish polynomial edge corrections via an algorithm with an independent interest in determining the optimal GCD cofactors with the lowest degree for arbitrary families of polynomials. Full article

Glaucoma is a leading cause of irreversible blindness globally, with early diagnosis being crucial to preventing vision loss. Traditional diagnostic methods, including fundus photography, OCT imaging, and perimetry, often fall short in sensitivity and fail to integrate structural and functional data. This study proposes a novel multi-modal diagnostic framework that combines convolutional neural networks (CNNs), vision transformers (ViTs), and quantum-enhanced layers to improve glaucoma detection accuracy and efficiency. The framework integrates fundus images, OCT scans, and clinical biomarkers, leveraging their complementary strengths through a weighted fusion mechanism. Datasets, including the GRAPE and other public and clinical sources, were used, ensuring diverse demographic representation and supporting generalizability. The model was trained and validated using cross-entropy loss, L2 regularization, and adaptive learning strategies, achieving an accuracy of 96%, sensitivity of 94%, and an AUC of 0.97—outperforming CNN-only and ViT-only approaches. Additionally, the quantum-enhanced architecture reduced computational complexity from O(n²) to O (log n), enabling real-time deployment with a 40% reduction in FLOPs. The proposed system addresses key limitations of previous methods in terms of computational cost, data integration, and interpretability. The proposed system addresses key limitations of previous methods in terms of computational cost, data integration, and interpretability. This framework offers a scalable and clinically viable tool for early glaucoma detection, supporting personalized care and improving diagnostic workflows in ophthalmology. Full article

Functional segregation in brain networks refers to the division of specialized cognitive functions across distinct regions, enabling efficient and dedicated information processing. This paper explores the significance of functional segregation in shaping brain network architecture, highlighting methodologies such as modularity and local efficiency that quantify the degree of specialization and intra-regional communication. We examine how these metrics reveal the presence of specialized modules underpinning various cognitive processes and behaviors and discuss the implications of disruptions in functional segregation in neurological and psychiatric disorders. Our findings underscore the fact that understanding functional segregation is crucial for elucidating normal brain function, identifying biomarkers, and developing therapeutic interventions. Overall, functional segregation is a fundamental principle governing brain organization, and ongoing research into its mechanisms promises to advance our comprehension of the brain’s complex architecture and its impact on human health. Full article

A two-dimensional time-dependent model is presented for upward-propagating internal gravity waves generated by an imposed thermal forcing in a layer of fluid with uniform background velocity and stable stratification under the anelastic approximation. The configuration studied is representative of a situation with deep or shallow latent heating in the lower atmosphere where the amplitude of the waves is small enough to allow linearization of the model equations. Approximate asymptotic time-dependent solutions, valid for late time, are obtained for the linearized equations in the form of an infinite series of terms involving Bessel functions. The asymptotic solution approaches a steady-amplitude state in the limit of infinite time. A weakly nonlinear analysis gives a description of the temporal evolution of the zonal mean flow velocity and temperature resulting from nonlinear interaction with the waves. The linear solutions show that there is a vertical variation of the wave amplitude which depends on the relative depth of the heating to the scale height of the atmosphere. This means that, from a weakly nonlinear perspective, there is a non-zero divergence of vertical momentum flux, and hence, a non-zero drag force, even in the absence of vertical shear in the background flow. Full article

As cities grow denser, the demand for efficient parking systems becomes more critical to reduce traffic congestion, fuel consumption, and environmental impact. This paper proposes a smart parking solution that combines deep learning and algorithmic sorting to identify the nearest available parking slot in real time. The system uses several pre-trained convolutional neural network (CNN) models—VGG16, ResNet50, Xception, LeNet, AlexNet, and MobileNet—along with a lightweight custom CNN architecture, all trained on a custom parking dataset. These models are integrated into a mobile application that allows users to view and request nearby parking spaces. A merge sort algorithm ranks available slots based on proximity to the user. The system is validated using benchmark datasets (CNR-EXT and PKLot), demonstrating high accuracy across diverse weather conditions. The proposed system shows how applied mathematical models and deep learning can improve urban mobility through intelligent infrastructure. Full article

Stokes wave is a classical problem in physics. Various Stokes wave theories in different forms have been developed to help us better understand their characteristics and for engineering applications. Exploring whether these Stokes wave theories can be converted into each other is a mathematical issue. We select three Stokes wave theories with different expansion parameters, all expressed in terms of water depth measured from the mean water level (MWL). Using series reversion to convert between the different expansions, we successfully transform the expressions for the velocity potential, wave profile, and dynamic properties between two of the Stokes wave theories. Through this conversion, we identify an incorrect expression for the water level in one Stokes wave theory. Full article

Psychological momentum dynamics in tennis have triggered interest for a long time, but measuring their impact presents substantial obstacles. In this paper, we present an approach to quantify momentum that combines real-time winning probabilities, leverage, and an exponentially weighted moving average (EWMA). We test the method on a high-profile match between Carlos Alcaraz and Novak Djokovic, demonstrating how changes in leverage affect momentum. Furthermore, we use feature extraction methods from time series analysis to derive momentum-related characteristics, which are critical inputs for creating an eXtreme Gradient Boosting (XGBoost) binary classification model to predict game winners. The algorithm has an average accuracy of 84% and provides real-time predictions of each player’s chances of winning the match. Our findings indicate that momentum is a somewhat relevant element in forecasting match outcomes, highlighting its potential value in improving match prediction systems. Full article

The rapid advancement of machine learning and deep learning techniques has revolutionized stock market prediction, providing innovative methods to analyze financial trends and market behavior. This review paper presents a comprehensive analysis of various machine learning and deep learning approaches utilized in stock market prediction, focusing on their methodologies, evaluation metrics, and datasets. Popular models such as LSTM, CNN, and SVM are examined, highlighting their strengths and limitations in predicting stock prices, volatility, and trends. Additionally, we address persistent challenges, including data quality and model interpretability, and explore emerging research directions to overcome these obstacles. This study aims to summarize the current state of research, provide insights into the effectiveness of predictive models. Full article