Research Articles (Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/159

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A robust fitted finite difference method for semi-linear two-parameter singularly perturbed pdes
(University of Guilan, 2026) Mohye, Mekashaw Ali; Munyakazi, Justin B.; Dinka, Tekle Gemechu; Hussen, Yusuf; Nura, Abe
This article presents a novel numerical scheme for solving nonlinear two-parameter singularly perturbed initial-boundary value problems. The proposed method combines an explicit forward Euler discretization for the temporal derivative with a fitted operator finite difference technique in the spatial domain. To handle the nonlinearity, Newton’s quasilinearization technique is employed. A comprehensive error analysis establishes that the developed scheme is first-order convergent uniformly in the perturbation parameters. The efficacy of the method is validated through two numerical examples, implemented in Python. The results, presented in tables and figures, demonstrate the method’s robustness and accuracy in resolving boundary layers for a wide range of perturbation parameters.
A second-order nonstandard finite difference method for a malaria propagation model with control
(Multidisciplinary Digital Publishing Institute (MDPI), 2026) Marime, Calisto B.; Munyakazi, Justin B.
Standard numerical methods such as Runge–Kutta and Euler methods have been widely used to approximate solutions to nonlinear systems. These methods converge to the solution only for small step sizes; for larger time steps, they generally generate spurious or chaotic solutions. In this paper, we consider a malaria propagation model with control for which we construct a second-order nonstandard finite difference scheme that preserves the important mathematical properties of the continuous model, which are positivity, boundedness, and stability of solutions irrespective of the step size. Moreover, we show that the equilibrium points of the discrete model are the same as those of the continuous model. By applying the double mesh principle, we provide evidence that the second-order NSFD scheme approximates the true solution with small errors. Theoretical assertions and numerical results show the advantages of the developed second-order nonstandard finite difference method.
Almost extraspecial structures and pseudofermionic operators
(Multidisciplinary Digital Publishing Institute (MDPI), 2026) Otera, Daniele Ettore; Russo, Francesco G.
We survey some recent combinatorial properties, which have been found in the context of the algebras of ladder operators in quantum mechanics. More specifically, we review dynamical systems which have nonselfadjoint Hamiltonians and are subject to a formalization in terms of pseudofermionic operators. For these systems, we detect structural analogies between algebras of pseudofermionic operators and the abstract notion of central product, which was originally studied for finite groups.
On the subdirect product of graph bundles
(Springer, 2026) Russo, Francesco G.; Bavuma, Yanga; Spessato, Stefano
The subdirect product of two finite groups A and B is defined as a subgroup of the direct product A × B. Although it is clear that, under appropriate choices of sets of generators S, SA, and SB, the Cayley graph Cay(A × B, S) corresponds to the Cartesian product Cay(A, SA)□Cay(B, SB) of two graphs, there is no analogue at the level of graph product that reflects the notion of subdirect product of groups. This is precisely the problem we discuss here. By using the concept of graph bundles and introducing the corresponding pullbacks we define an operation on graph bundles such that the Cayley graph of the subdirect product of two groups can be described as the total space of the product of the Cayley graphs. This allows us to define the so-called “network K-theory group of a graph”, inspired by the notion of topological K-theory, and we are able to investigate an interesting functor from the category of graphs to the category of Abelian groups.
Physics-Informed Neural Network approach to time-fractional Black–Scholes models: Pricing down-and-in Parisian options under rough volatility
(Elsevier B.V., 2026) Patidar, Kailash; Tarla, Divine; Nuugulu, Samuel
This paper investigates the pricing of down-and-in Parisian options within a time-fractional rough volatility Black–Scholes framework. The proposed model incorporates both fractional dynamics and rough volatility effects, allowing for the simultaneous representation of long-range dependence, memory effects, and highly irregular volatility paths observed in financial markets. The fractional component introduces non-local temporal behaviour, capturing persistence and subdiffusive dynamics, while the rough volatility structure, driven by a Volterra-type process with Hurst parameter H<0.5, accounts for the observed anti-persistent and oscillatory nature of volatility. The resulting time-fractional rough volatility Black–Scholes partial differential equation (tf-RVBSPDE) extends classical models by embedding both path dependence and global memory into the pricing mechanism. In addition, the Parisian feature is formulated through an occupation-time constraint, ensuring that barrier activation depends on sustained excursions rather than instantaneous crossings, thereby enhancing stability in volatile market conditions. To address the analytical and computational challenges posed by the non-local and non-Markovian structure of the model, a Physics-Informed Neural Network (PINN) approach is employed. The Grünwald–Letnikov formulation of the fractional derivative is adopted to ensure compatibility with automatic differentiation and efficient numerical implementation. The PINN method approximates the solution by minimizing a loss function that enforces both the governing equation and boundary conditions. Numerical experiments based on S&P 500 data demonstrate that the proposed approach accurately captures the effects of rough volatility and memory, with improved performance in regimes where α<0.5. The results confirm that the model recovers classical solutions in limiting cases and provides a robust pricing approach for cases when tractable analytical solutions are non-existing. Overall, the approach establishes a unified connection between fractional stochastic dynamics, rough volatility modelling, and PINN based numerical methods for pricing complex path-dependent derivative contracts.
Mapping breast cancer protein interaction networks as metric spaces: insights into central zones and drug discovery targets
(AccScience Publishing, 2025) Fadhal, Emad
Introduction: Graph theory was employed in recent advances of cancer research for gain deeper insights into the complex structure and function of protein-protein interaction (PPI) networks. Objective: By representing proteins as nodes and their interactions as edges, graph theory offers a comprehensive framework for analyzing the topological properties of these networks and identifying key nodes that regulate critical biological processes. This approach has been widely applied to study various cancers, including breast cancer. Methods: To investigate the molecular organization and critical pathways in breast cancer, we constructed a breast cancer protein-protein interaction network (BCPIN) and analyzed its hierarchical structure. The network was modeled as a metric space to delineate its central zones, facilitating the identification of essential hubs enriched with signaling pathways critical for cancer progression. Results: Our study demonstrates the potential of hierarchical modeling of the BCPIN in unraveling its molecular organization and identifying therapeutic opportunities. By analyzing PPI network as a metric space, we highlight central zones 1 – 3 as critical hubs enriched with key signaling pathways, such as DNA repair, Notch signaling, and p53 signaling, which are essential to cancer progression. The identification of MAPK14 as a central node emphasizes its significant role in cancer biology and its value as a therapeutic target. The predominance of signaling proteins within these zones underscores their functional relevance, offering a strong rationale for prioritizing them in drug development. Conclusion: By modeling the PPI network as a metric space, we uncovered important insights into its architecture and the central zone’s critical role in facilitating key cellular processes. Our results indicate that zones 1 – 3, particularly the central zone, may serve as promising targets for drug discovery in cancer biology.
Physics-informed neural network approach to time-fractional black–scholes models: pricing down-and-in parisian options under rough volatility
(Elsevier B.V., 2026) Nuugulu, Samuel Megameno; Patidar, Kailash; Divine Tawe Tarla
This paper investigates the pricing of down-and-in Parisian options within a time-fractional rough volatility Black–Scholes framework. The proposed model incorporates both fractional dynamics and rough volatility effects, allowing for the simultaneous representation of long-range dependence, memory effects, and highly irregular volatility paths observed in financial markets. The fractional component introduces non-local temporal behaviour, capturing persistence and subdiffusive dynamics, while the rough volatility structure, driven by a Volterra-type process with Hurst parameter H<0.5, accounts for the observed anti-persistent and oscillatory nature of volatility. The resulting time-fractional rough volatility Black–Scholes partial differential equation (tf-RVBSPDE) extends classical models by embedding both path dependence and global memory into the pricing mechanism. In addition, the Parisian feature is formulated through an occupation-time constraint, ensuring that barrier activation depends on sustained excursions rather than instantaneous crossings, thereby enhancing stability in volatile market conditions. To address the analytical and computational challenges posed by the non-local and non-Markovian structure of the model, a Physics-Informed Neural Network (PINN) approach is employed. The Grünwald–Letnikov formulation of the fractional derivative is adopted to ensure compatibility with automatic differentiation and efficient numerical implementation. The PINN method approximates the solution by minimizing a loss function that enforces both the governing equation and boundary conditions. Numerical experiments based on S&P 500 data demonstrate that the proposed approach accurately captures the effects of rough volatility and memory, with improved performance in regimes where α<0.5. The results confirm that the model recovers classical solutions in limiting cases and provides a robust pricing approach for cases when tractable analytical solutions are non-existing. Overall, the approach establishes a unified connection between fractional stochastic dynamics, rough volatility modelling, and PINN based numerical methods for pricing complex path-dependent derivative contracts.
A class of unitary modules with stable range 2
(Palestine Polytechnic University, 2025) Ali, Abdelhamid M.A; Mehdi-Nezhad, Elham; Rahimi, Amir M
The notion and some properties of B-rings, in a natural way, are extended to B-stable and BJ-stable modules. Let J(R) be the Jacobson radical of a commutative ring R with identity 1 ≠ 0. A sequence (a1, …, an+1), n ≥ 1, of elements in R is said to be unimodular if 1 (Formula present). A ring R is said to be a B-ring if for any unimodular sequence (Formula present), there exists an element b in R such that (Formula present). Our aim in this paper is to extend this notion in the context of modules. It is shown that a cyclic module over a B-ring (in particular, semilocal ring, Noetherian ring in which every prime ideal is maximal, Dedekind domain) is BJ-stable. Also, a torsion-free cyclic R-module is B-stable if and only if R is a B-stable ring. Any module of rank greater than or equal to 3 is never BJ-stable and consequently not B-stable. We show that a finitely generated R-module A is BJ-stable if Z(B) is finite for every submodule B of A with B ⊈ J(A), where Z(B) denotes the set of all maximal submodules of A containing B.
A second-order nonstandard finite difference method for a malaria propagation model with control
(Multidisciplinary Digital Publishing Institute (MDPI), 2026) Marime Calisto Blessmore; Munyakazi, Justin B.
Standard numerical methods such as Runge–Kutta and Euler methods have been widely used to approximate solutions to nonlinear systems. These methods converge to the solution only for small step sizes; for larger time steps, they generally generate spurious or chaotic solutions. In this paper, we consider a malaria propagation model with control for which we construct a second-order nonstandard finite difference scheme that preserves the important mathematical properties of the continuous model, which are positivity, boundedness, and stability of solutions irrespective of the step size. Moreover, we show that the equilibrium points of the discrete model are the same as those of the continuous model. By applying the double mesh principle, we provide evidence that the second-order NSFD scheme approximates the true solution with small errors. Theoretical assertions and numerical results show the advantages of the developed second-order nonstandard finite difference method.
Mathematical modelling of malaria spread in response to climate variability in Sudan
(Payame Noor University, 2025) Farah, Gassan; Mukhtar, Abdulaziz; Patidar, Kailash
Malaria continues to represent a significant public health concern in Sudan‎, ‎with cases rising over 40% from 2015 to 2020‎. ‎This research investigates how climate change affects malaria transmission patterns using a mathematical model in an ordinary differential equation framework‎. ‎The analysis involves calculating the basic reproduction number and evaluating the system's qualitative properties to gain insights into disease dynamics‎. ‎Additionally‎, ‎a sensitivity analysis is conducted to evaluate how climatic conditions‎, ‎e.g.‎, ‎rainfall and temperature, influence key model parameters‎. ‎Statistical approaches are utilized to estimate parameters‎ ‎and calibrate the model using empirical data from Sudan‎, ‎ensuring consistency between the model and observed trends‎. ‎Numerical simulations demonstrate the growing influence of climate variability on the spatial distribution of malaria vectors and the transmission progression over time‎. ‎The study establishes a strong association between climatic changes and the exacerbation of malaria prevalence in Sudan‎. ‎These findings emphasize the urgent need for climate-adaptive strategies‎, ‎including improved vector control‎, ‎strengthened surveillance systems‎, ‎and climate-resilient public health interventions‎, ‎to address the increased risks posed by changing environmental conditions‎. ‎The research provides valuable insights to inform evidence-based policies aimed at reducing malaria transmission in Sudan and other regions that are experiencing similar challenges due to climate change‎.
McKinsey-Tarski algebras and raney extensions
(Springer Science and Business Media B.V., 2026) Suarez, Anna Laura; Bezhanishvili, G.; Raviprakash R.
We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the T0-hull of a Raney extension generalizes that of the TD-hull of a frame.
Nilpotent locally compact groups with small topological entropy
(Springer Science and Business Media BV, 2026) Russo, Francesco G; Waka, Olwethu
We characterize the finiteness of the topological entropy of continuous automorphisms of locally compact nilpotent p-groups (p prime) via the notion of p-rank. Considering upper unitriangular matrices over the p-adic integers and p-adic rationals, we present an algorithmic criterion in order to produce nilpotent locally compact p-groups of large nilpotency class and with continuous automorphisms of finite topological entropy. The procedure allows us to generalize the construction of large families of totally disconnected locally compact Heisenberg p-groups. It should be also mentioned that alternative arguments have been proposed, in order to avoid the use of the p-rank for the finiteness of the topological entropy of the continuous automorphisms, but these arguments involve the notion of topologically capable group, which wasn't explored for locally compact groups (except for the discrete case).
Hereditary interior operators
(Springer Science and Business Media Deutschland GmbH, 2026) Assfaw, Fikreyohans Solomon; Holgate, David
Working in an arbitrary category endowed with a fixed -factorization system in which the preimage functor for any given morphism preserves arbitrary joins we improve the Castellini notion of hereditary interior operators. We introduce and study a general notion of hereditary interior operators in terms of dual images. It is shown that these operators behave as well as hereditary closure operators. In particular, hereditary interior operators are characterized in terms of the notion of initiality as for hereditary closure operators. Moreover, we prove that additive hereditary interior operators are Castellini’s hereditary interior operators. Some examples are included.
AI-powered platforms in STEM education, insights from UTAUT model: PLS-SEM and artificial neural networks hybrid analysis
(Elsevier B.V., 2025) Bayaga, Anass
This investigation examines the effect of AI-powered technologies on STEM cognition by means of the unified theory of acceptance and use of technology (UTAUT) model. A sample of 160 respondents from diverse academic backgrounds was analysed by partial least squares structural equation modelling (PLS-SEM) to test the UTAUT model. Furthermore, artificial neural networks with a multilayer perceptron (MLP) architecture were employed to validate the outcomes and forecast the relationships amongst the UTAUT constructs. The examination focused on the core UTAUT constructs: performance expectancy, EE, SI, facilitating conditions, and their influence on the behavioural intention and actual use of AI tools in STEM education. The outcomes indicated that performance expectancy and facilitating conditions are significant predictors of behavioural intention (BI), signifying the perceived usefulness of AI tools and the existence of supportive didactic settings are critical for their adoption. Nevertheless, the conjectured moderating effects of user gender and institutional role were not strongly supported. However, the hypothesized moderating effects of user gender and institutional role on these relationships were not strongly supported, inferring a more universal applicability of AI technologies across demographic sets. These outcomes highlight the necessity of integrating AI tools into educational practices in a way that is collectively available and operative, nurturing wide-ranging educational environment that aids all users irrespective of demographic variances.
An efficient physics-informed neural network solution to the time-space fractional black-scholes equation
(Springer International Publishing, 2025) Tarla, Divine T.; Patidar, Kailash C; Nuugulu, Samuel M.
This study develops a rigorous analytical and computational framework for solving the time-space-fractional Black–Scholes equation (ts-fBSE), a generalization of the classical Black–Scholes model that captures nonlocal temporal memory and spatial anomalous diffusion in financial markets. Starting from fractional stochastic dynamics driven by Gaussian white noise, we derive the ts-fBSE using generalized Itô–Lévy calculus and establish its well-posedness under appropriate initial and boundary conditions. We demonstrate that the conventional transformation y=lnS+a does not, in general, reduce the spatial operator to integer order and provide an alternative transformation that yields a constant-coefficient time-fractional BSPDE. The equation is solved using a physics-informed neural network (PINN) incorporating the Grünwald–Letnikov fractional derivative through a stable matrix formulation, eliminating mesh discretization and stability constraints typical of finite-difference methods. The PINN loss functional enforces the operator residual in L2(Ω) augmented by boundary and terminal penalties, trained with a piecewise-constant decay learning rate and a stopping tolerance of 10-4. Numerical experiments for European put options validate the accuracy and stability of the method, showing decreasing mean absolute error as the fractional order α→1. The results confirm that the proposed PINN framework provides a mathematically consistent and computationally robust alternative for solving fractional–stochastic PDEs in quantitative finance, complementing recent developments such as fPINNs and XPINNs.
Geometrical representation and Hirota direct approach for multiple soliton solutions of nonlinear M-coupled fractional equations
(Taylor and Francis Ltd., 2025) Farah, Gassan A.M.O.; Abakar Abdalla Hassaballa; Abdel-Salam, Emad A.B.
This paper introduces a novel analytical framework for deriving multiple soliton and singular soliton solutions to M-coupled fractional evolution equations. By integrating conformable fractional derivatives with an extended Hirota direct method, we systematically solve fractional versions of the KdV, mKdV, KP, and modified KP equations. The conformable derivative permits effective bilinearization, facilitating the construction of explicit solutions. We further provide a geometric interpretation through curvature analysis of soliton surfaces in fractional space. Theoretical results are validated against classical cases (α = 1), demonstrating consistency and enhancing the analytical toolkit for modeling wave propagation in nonlinear optics, plasma physics, and anomalous diffusion.
On Császár structures and pre-nearness on frames
(Springer Science and Business Media Deutschland GmbH, 2026) Holgate, David; Iragi, Bakulikira
The aim of this paper is to introduce the concept of semi-Császár structures and investigate their relationship with the well-known notion of pre-nearness structures on frames. More explicitly, we define the category of semi-Császár frames and establish a connection with the category of covering pre-nearness frames. We provide conditions under which semi-Császár structures relate with pre-uniformities on frames. Finally, we present a frame counterpart to the relationship between symmetric syntopogenous structures and nearness spaces as established by Herrlich (Gen Topol Appl 4(3):191–212, 1974).
Mathematical modeling of influence of multiplicative white noise on dynamical soliton solutions in the KdV equation
(World Scientific, 2025) Farah, Gassan; Hassaballa, Abaker; Yavuz, Mehmet
This study investigates the impact of multiplicative white noise on soliton solutions of the Korteweg–de Vries (KdV) equation, a classical model for nonlinear wave propagation. While the deterministic KdV framework effectively captures soliton dynamics in idealized settings, real-world systems often experience stochastic fluctuations that can significantly affect wave behavior. By introducing space–time-dependent multiplicative noise, we formulate a stochastic KdV (SKdV) equation and derive analytical expressions for both single and multi-soliton solutions using a novel wave transformation in conjunction with the Hirota bilinear method. Numerical simulations across varying noise intensities reveal that stochastic perturbations lead to soliton deformation, amplitude modulation, phase shifts, and disrupted coherence. These effects become more pronounced with stronger noise, altering soliton interactions and long-term stability. The findings emphasize the importance of incorporating stochastic effects into non-linear wave models, with implications for applications in fluid dynamics, plasma physics, and optical communications.
A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems
(Springer Nature, 2025) Munyakazi, Justin B; Mohye, Mekashaw Ali; Dinka, Tekle Gemechu
This article focuses on the numerical solution of a time-dependent parabolic problem that exhibits singular perturbations and involves two perturbation parameters. To address this problem, a fitted mesh finite difference method is developed. In numerical discretization, the implicit Crank-Nicolson technique is employed to discretize the time derivative using a uniform mesh. As for the spatial derivative, a hybrid finite difference scheme known as the adaptive fitted mesh of the Shishkin type is utilized. The study also includes a discussion on a priori bounds for the continuous solution and its derivatives. The proposed method is proven to be uniformly convergent of order two in both time and space. Theoretical analysis and simulations on various test examples confirm the scheme’s accuracy and convergence properties
Network topology similarities across cancer types: identifying central protein hubs for drug discovery
(LIDSEN Publishing Inc, 2025) Fadhal, Emad
A molecular-level understanding of cancer is essential for the development of effective therapies. Constructing protein-protein interaction (PPI) networks offers a valuable approach to identifying dysregulated driver genes and potential therapeutic targets. In this study, we modeled cancer PPI networks as metric spaces and applied mathematical and computational algorithms to analyze their structural and functional properties. Our findings reveal that these networks share a conserved architecture across different cancer types, with central zones enriched in essential proteins and critical regulatory pathways. Notably, zones 1 and 2 of the cancer PPI networks are uniquely enriched in specific pathways, underscoring their importance in the progression of cancer. These results highlight the potential of metric-based analysis of PPI networks to uncover key molecular targets and accelerate drug discovery in oncology

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