Punjab University Journal of Mathematics
Punjab University Journal of Mathematics (PUJM) is a distinguished publication of the Institute of Mathematics, University of the Punjab, Lahore. It is indexed in the Master Journal List – Emerging Sources Citation Index (ESCI). In the Journal Citation Reports by Thomson Reuters 2023, PUJM achieved an impressive impact factor of 1.0, reflecting its rising prominence and scholarly contribution in the field of mathematical sciences. The journal is published on a monthly basis and is dedicated to disseminating high-quality research in mathematics. The official abbreviated title, as assigned by the ISSN authority, is Punjab Univ. J. Math. The first issue of the journal was published in 1967 by Prof. Dr. Syed Manzur Hussain (Late). Article Processing Charges: It is pertinent to mention that from January 2025 the following Article Processing Charges (APC) will be implemented on every article accepted for publication in PUJM. For International Authors: $100 For Local Authors: Rs.15000/- per paper For Local and Foreign Joint Authorship: $100 Prof. Dr. Ghazala Akram, PhD & PostDoc (Editor in Chief) Institute of Mathematics, University of the Punjab, Lahore, Pakistan. Dr. Saima Arshed, PhD (Managing Editor) Institute of Mathematics, University of the Punjab, Lahore, Pakistan.
Some Efficient Derivative-Based Semi-Open Quadrature Rules with Error Analysis
This study suggests four innovative and effective quadrature techniques that combine functional evaluations and their first-order derivatives for data points that are identically spaced, with an emphasis on computational efficiency regarding time and cost utilisation. All the techniques are theoretically derived, and the theorems concerning accuracy,precision, and error terms are also established. The suggested approaches are derivative-based semi-open-type rules. Compared to the conventional rules, the proposed methods are more accurate and possess higher precision degree. Several numerical experiments are conducted to compare the accuracy, truncation errors, rates of convergence, cost evaluation and average execution times of the new approaches compared with the conventional methods. Because of their promisingly lower computational costs, the results of the analysis demonstrate that the developed methods are more efficient than the original methods from both a theoretical and numerical aspects.
Semi-open Newton-Cotes formulas, Derivative-based, Precision, Accuracy, Time and cost effectiveness.
Copyright (c) 2025 Mahesar et al. https://creativecommons.org/licenses/by-nc-nd/4.0
QSPR Analysis of Antiplatelet Agents and Dual Antiplatelet Therapy Using Labeled-Based Topological Indices and Lagrange Interpolation Techniques
This research presents a comprehensive approach to the quantitative structure-property relationship (QSPR) analysis of antiplatelet agents (ATA) Dual Antiplatelet Therapy (DAPT) drugs, including aspirin, prasugrel, clopidogrel, and ticagrelor, by employing labeled-based topological indices. In chemical graph theory, the molecular structure of these drugs was encoded into graph representations; their features were captured by the labeled topological indices calculated in the molecular architecture. These indices are then used to predict critical physicochemical properties like boiling point, molar refractivity, and enthalpy of vaporization. To further improve the accuracy of property prediction, we introduce the Lagrange interpolation polynomial as a powerful curve-fitting technique and demonstrate its superiority over traditional regression models. The results emphasize the robustness and reliability of label-based topological indices in QSPR analysis, providing new insights into drug design and molecular characterization. This methodology not only enhances our understanding of the physicochemical properties of DAPT drugs but also opens doors to future advancements in computational chemistry, especially in drug development and optimization
Labeled Topological indices, QSPR analysis, Antiplatelet agents, Dual Antiplatelet therapy, Lagrange Interpolation Polynomial.
Copyright (c) 2025 Virk et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Optimizing decision making for sustainable solar thermal energy systems within an intuitionistic fuzzy environment
In the context of addressing global energy needs and climate change, the transition to sustainable energy systems has become one of the most important objectives, and in this context. Solar-thermal energy systems are among the most important sources of renewable energy solu tions. However, existing research on solar-thermal energy systems often falls short in effectively handling uncertainty and the credibility of expert evaluations, which are essential for ensuring reliable decision-making. Traditional methods still focus on deterministic or simplified models, ne glecting the intricate and uncertain evaluations present in real-world sit uations. To address these shortcomings, this study employs intuitionistic fuzzy credibility numbers (IFCNs), which explicitly incorporate member ship, non-membership, and credibility degrees, enabling a more compre hensive representation of uncertain and hesitant information. Based on IFCNs, we introduce a new, more effective method of handling credibil ity in aggregation named the generalized intuitionistic fuzzy credibility numbers weighted averaging (GIFCNWA) operator. Based on this opera tor, a multi-attribute decision-making algorithm is proposed that integrates attribute importance and reliability of evaluations. The proposed method ology is applied to a real-world decision-making problem in solar ther mal energy storage by considering four options: nanostructured absorber coatings, advanced phase-change materials, high-temperature alloys, and adaptive control systems. The results identify nanostructured absorber coatings as the most optimal solution. A parametric sensitivity analysis is conducted, demonstrating the robustness and stability of the proposed op erator under varying parameter conditions. A comparative analysis with existing decision-making techniques reveals the superiority and consis tency of the proposed operator in delivering reliable results.
Intuitionistic fuzzy set, Credibility numbers, Generalized intuitionistic fuzzy credibility numbers weighted averaging operator, Multi-attribute decision-making, Sustain able energy systems, Solar energy
Copyright (c) 2025 Anjum et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Neural Computing Approach to the Flow and Thermal Analysis of Ternary Hybrid Nanofluid Over Wedges
The steady state motion of a ternary hybrid nanofluid across a permeable wedge is investigated in the case when the wedge is stretching or contracting. An applied magnetic field and the effect of radiation are included in the analysis. An application of a similarity transformation to the governing equations that describe the flow of ternary hybrid nanofluid results in the conversion of those equations into ordinary differential equa tions. All of these equations are solved numerically using the bvp4c solver in MATLAB. Additionally, the flow of the ternary hybrid nanofluid is governed by the nonlinear differential equations that are then solved by a hybrid approach combining artificial neural networks with Levenberg Marquardt backpropagation technique. The performance and hence the comparison of algorithms are carried out using regression analysis, er ror histograms, function fitting graphs, and mean squared error results. To achieve the desired results, the considered parameters are varied systematically to be the stretching/shrinking parameter (λ), the magnetic parameter (M), the suction/injection parameter (S), the wedge angle (m), and the radiation parameter (Rd). In terms of performance, the velocity and tem perature profiles for the proposed model demonstrated strong accuracy, as evidenced by the mean squared error values of [2.251 × 10−9, 5.5339 × 10−10, 1.2449 × 10−9, 1.7511 × 10−8, 3.1377 × 10−9]. According to the investigation, solutions are found up to a particular suction and stretching/shrinking strength. An increase in the wedge angle parameter, the critical value associated with the thermal response decreases. Additionally, the heat transfer rate provided by the ternary hybrid nanofluid is greater than those possessed by the conventional nanofluids. Moreover, an increase in the radiation parameter enhances the heat transferring rate.
Ternary hybrid nanofluid, Magnetic field, Stretching/shrinking wedge, Artifi cial neural networks, Similarity transformation.
Copyright (c) 2025 Asjad et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Application of Cubic B-Spline Functions in Galerkin Finite Element Method for Solving Second Order Sub-Diffusion Equation
In this study, the Galerkin finite element method (FEM) is applied to find the numerical solutions of the second-order sub-diffusion equation. The proposed approach employs cubic B-spline functions as both the trial and test functions. The weak formulation of the governing equation is developed, and the connection between the global and local coordinate systems is established through a suitable transformation. For time discretization, the standard finite difference formulation is used for the time derivative, while the Crank-Nicolson scheme is applied to approximate the unknown functions. A stability analysis is performed to verify the robustness of the scheme and to ensure that no numerical errors grow during computations. Additionally, error estimates are derived to assess the accuracy of the proposed method. The efficiency and reliability of the developed scheme are demonstrated by solving several test problems. The computed results are presented in both graphical and tabular forms, showing good agreement with the exact or reference solutions.
Diffusion equation, Galerkin method, Cubic B-spline basis functions, Finite element method.
Copyright (c) 2026 Iqbal et al. https://creativecommons.org/licenses/by-nc-nd/4.0
A class of non-stationary ternary 4-point subdivision schemes based on iterations
This paper presents a novel class of nonstationary ternary 4-point subdivision schemes, capable of generating C³ continuous limit curves. These schemes are built upon the generalized ternary scheme of order 5 presented in [19] and iterated functions. A distinctive feature is that the shape of the resulting limit curve varies with alterations in the initial parameters. To facilitate localized shaping adjustments, we devise a non-uniform subdivision scheme that extends our non-stationary schemes. This method allows for Tension parameters are to be assigned to each edge of the initial control polygon, offering enhanced flexibility and precision in curve shaping.
Non-stationary ternary scheme; Smoothness; Iteration; Non-uniform
Copyright (c) 2025 Zhang & Zheng https://creativecommons.org/licenses/by-nc-nd/4.0
On Novel Extension of n-Polynomial Pre-invex Functions and Related Post-Quantum Integral Inequalities with Application
In this paper, we introduce a new class of n-polynomial pre-invex functions to investigate the refined bounds of Hermite-Hadamard type inequalities involving post-quantum integrals. We establish an advanced and comprehensive multi-parameter identity and several post-quantum integrals to aid our main results. By applying this generic identity, we estimate the unique and sharp bounds for the functions, which are bounded and satisfy the Lipschitz condition. In order to find the optimal bounds, we examine these results for the best choice of parameters involved and express those graphically to validate our results. In the end, we calculate certain special means as the application of presented results.
Pre-invex function;Multi parameters;n-polynomial pre-invexity;Power mean inequality;Lipschitz condition
Copyright (c) 2025 Nawaz & Shahzad https://creativecommons.org/licenses/by-nc-nd/4.0
Energy of Fuzzy Hypersoft Sets with Application in Machine Learning for Decision Making
This study advances the application of fuzzy hypersoft sets (FHSS)in machinelearning (ML) decision-making by introducing a novel energy metric to quantify multi-sub-attribute uncertainty. Building on fuzzy set theory, soft sets, and hypersoft sets, FHSS integrates fuzzy membership with multi-sub-attribute parameterization, addressing limitations of traditional uncertainty models in handling complex, high-dimensional datasets. Inspired by spectral graph theory, the proposed energy metric—the sum of singular values of the FHSS matrix — quantifies systemic significance and enables a robust ranking of alternatives. An algorithm leveraging this metric has been developed and validated through appli cations in healthcare (heart risk profiling) and energy systems, achieving 90.83% accuracy and an F1-score of 0.8706 in a dataset of 500 participants. A comparative analysis demonstrates the superiority of FHSS energy over fuzzy soft and hypersoft sets, particularly in capturing attribute interdependencies. Despite the computational challenges posed by large matrices, the framework provides interpretable and scalable solutions for ML-driven decision-making under uncertainty. Future work will optimize computational efficiency and extend applications to domains such as financial risk analysis, further reinforcing FHSS energy as a transformative tool for precise, uncertainty-aware decision-making.
Fuzzy Hypersoft Set, Energy Metric, Machine Learning, Decision-Making, Healthcare Diagnostics, Singular Value Decomposition, Fuzzy Logic.
Copyright (c) 2025 Saeed et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Analytical solutions of time-fractional non-linear model Clannish Random Walker’s Parabolic equation and its sensitivity
In this article, the exact travelling wave solutions for the non linear time-fractional Clannish Random Walker’s Parabolic equation are discussed. This study establishes the new extended direct algebraic method in which periodic, bright, multiple U-shaped bright and kink-type wave solitons are obtained with exact solutions offered by the mixed hyperbolic and trigonometry solutions, mixed periodic and periodic solutions, plane solutions, shock solutions, mixed trigonometric solutions, mixed singular solutions, mixed shock single solutions, complex solitary singular solutions, shock solutions and shock wave solutions. The obtained solutions of the non-linear time-fractional Clannish Random Walker’s Parabolic equation model are graphically presented for different values of the involved parameters using Wolfram Mathematica software. The propagating behaviours are visualised through 3D, contour, and 2D surface plots to illustrate the influence of key parameters on the solution profiles. The time-fractional derivative introduces a memory effect into the model, making it more suitable for describing real-world physical processes that in volve hereditary and nonlocal behaviour. The presence of novel soliton structures, such as multiple U-shaped solitons and bright-shaped solitons, further highlights the novelty and complexity of the model’s dynamics. In this study, a planar dynamical system is constructed from the pro posed model to carry out a sensitivity analysis of initial conditions. This transformation enables the investigation of how small variations in the initial values influence the system’s long-term behaviour. The proposed method proves to be efficient, reliable, and broadly applicable for generating new analytical solutions to both integer and non-integer-order differential equations arising in mathematical physics and engineering.
New extended direct algebraic method; Time-fractional Clannish Random Walker’s Parabolic equation; Exact solution.
Copyright (c) 2025 Majid et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Advancements in Fuzzy Research in Pakistan: A Bibliometric Perspective between 1989 and 2023
Pakistan, as a developing nation, has contributed to fuzzy research through its researchers, whose work reflects a growing interest in both the theoretical and applied aspects. However, the extent and impact of these contributions remain unexplored in scholarly evaluations. Thus, this study aims to provide a bibliometric analysis of fuzzy research in Pakistan from 1989–2023, utilizing the Web of Science (WoS) core collection database and VOS viewer software. The key bibliometric indicators, including h-index, co-citation analysis, bibliographic coupling, and keyword co-occurrence, have been evaluated. The main findings show a significant growth, with total publications increasing from 346 (1989–2014) to 2,609 (2019–2023). Early research focused on foundational concepts like fuzzy sets and fuzzy logic, while recent studies emphasize applied domains such as decision-making, artificial intelligence, and machine learning. Leading contributors include Muhammad Akram (349 publications, 10,224 citations) and Tahir Mahmood (217 publications, 6,508 citations), with the University of the Punjab (PU) and COMSATS University Islamabad (CUI) emerging as the most productive institutions. Collaboration analysis identifies Saudi Arabia as the primary partner, contributing 50% of co-authored works, and Dragan Pamucar, a prominent collaborator from the University of Belgrade, Serbia. The topical and keyword analysis reveals the focus of the research area and its alignment with global trends in fuzzy theories, as well as a significant contribution across diverse disciplines. Subject areas like Business Economics, Optics, Robotics, Food Sciences, Biochemistry, and Social Sciences have fewer contributors and appear less aligned with global research trends. Institutions can encourage strategic publishing to improve global rankings and citation impact.
Bibliometrics; Web of Science; Scopus; fuzzy; VOSviewer.
Copyright (c) 2025 Saqlain et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Bounds on F-Index of Tricyclic Graphs with AI Applications
Topological indices, symmetric functions in graph theory, are critical tools for characterizing the structural and topological properties of molecules and networks. The Forgotten index (F-index), defined as the sum of the cubes of vertex degrees in a molecular graph, was intro duced by Furtula and Gutman to model the structural dependence of total π-electron energy. Tricyclic graphs, connected graphs of order m and size m+2, are of particular interest due to their intermediate complexity between bicyclic and tetracyclic structures. In this work, we establish sharp upper and lower bounds for the Forgotten index within the class of tri cyclic graphs and identify the extremal graphs achieving these bounds. To extend the theoretical contributions, we also propose to integrate artificial intelligence (AI) and machine learning (ML) methodologies, for their utility in three key areas: (1) Predictive modeling via graph neural networks (GNNs) to estimate F-indices for large-scale tricyclic graphs, bypassing combinatorial complexity; (2) Generative design using variational autoencoders (VAEs) to synthesize novel tricyclic graphs with near-optimal F indices for materials science applications; and (3) AI-driven optimization employing reinforcement learning (RL) to validate extremal graph structures and explore uncharted regions of the tricyclic graph space. We combine mathematical and AI approaches to improve F-index analysis in tricyclic graphs.
Tricyclic graphs; Graph transformations; Forgotten chemical invariant.
Copyright (c) 2025 Kanwal et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Perturbation and Neural Network Approximation for flexible Blade Coating Analysis of Third Order Fluid with Lubrication Approximation Theory
The flexible blade coating process is widely used in various in-dustries for the production of thin films on substrates. However, the analy-sis of coating process for non Newtonian fluids remains a challenging task due to their complex rheological behavior. In this work, we have used third order fluid along with flexible blade which is new and challenging because of the complexity of the non-linear equations, the surface being coated, decomposition of the functional material layers over complex geometries. The importance of our analysis is that it enables the optimization of the in-volved parameters which include gap height, blade angles, speed and most importantly fluid rheology. The perturbation approximation method with Levenberg–Marquardt neural network (LM–NN) is used to investigate the coating process of third order-fluids using lubrication approximation the-ory. The deformation of the blade during the coating process and behavior of fluid flow is represented in terms of system of nonlinear equations. Numerical solution has been obtained to analyze the blade flexibility and lubrication force on the coating process. For data training of LM–NN, 70 percent data is taken for training, 15 percent for validation and 15 per-cent data is used for testing. Our results show that the proposed method can accurately predict the coating thickness and the blade deformation for different values of the coating parameters. Additionally, various physical parameters which influence the fluid motion like flexibility and speed of blade, including its rheological characteristics, are graphically analyzed. The findings of the study suggest that a better understanding of the com-plex process of blade coating for non-Newtonian fluids can be explained through perturbation methods along with neural networks approximation, which can improve the coating process applications on industrial level.
Blade coating analysis, perturbation approximation with Neural networks, lubrication approximation theory, third order fluid.
Copyright (c) 2025 Bhatti et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Innovative Neural Network Approach for Solving TDKS Equations with Jensen’s Inequality
We propose a novel neural network-based approach for solv-ing the Time Dependent Kohn-Sham (TDKS) equations, central to Time-Dependent Density Functional Theory (TDDFT). Focusing on the one-electron case, where the TDKS reduces to the time-dependent Schr¨odinger equation, we employ modified Physics Informed Neural Networks (PINNs) incorporating Jensen’s inequality in place of the traditional Mean Squared Error (MSE) loss. This convex formulation improves training stability and solution accuracy. Compared to the classical Runge-Kutta (RK4) method, our approach achieves comparable accuracy while demonstrating supe-rior scalability and smoother convergence, especially for stiff or nonlinear dynamics. This work establishes a foundation for extending neural PDE solvers to more complex quantum systems.that the classroom teaching ex-periment began.
Jensen’s inequality . Mean squared error . Runge-Kutta methods . Kohn-sham equations
Copyright (c) 2026 Farooq et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Assessing Environmental Pollution: Quality Control Using CRITIC-WASPAS Methods under Intuitionistic Fuzzy Z-Numbers
Industrial pollution is still a major worldwide issue. Strong decision-making frameworks are required to evaluate and reduce its detri mental effects on the environment. An extensive case study assessing dif ferent industrial pollution produced by the energy industry, textile indus try, chemical industry, manufacturing industry, agriculture industry, and construction industry and its environmental impact is presented in this pa per. We present a structured evaluation of environmental degradation by analyzing key environmental criteria, including waste generation, noise pollution, soil contamination, air pollution, and water pollution. In or der to accomplish a systematic and objective evaluation, we provide an advanced multi-criteria decision-making (MCDM) framework that com bines the CRITIC-WASPAS and CRITIC-EDAS approaches with intu itionistic fuzzy Z-number (IFZN). This integration ensures a more reli able, flexible, and data-driven assessment of industrial pollution sources by addressing uncertainty, imprecision, and inconsistency in the data ef fectively. In order to determine the objective weights of environmental criteria , the CRITIC (Criteria Importance Through Intercriteria Correla tion) method is applied here. These weights undergo additional processing using the WASPAS and EDAS approaches, which combine aggregated sum product weighting (WASPAS) and positive and negative distance based evaluation (EDAS) to check the ranking of given criteria. Addition ally, sensitivity analysis is used to assess the robustness and dependabil ity of the suggested fuzzy model by altering input parameters, including weights, membership values, and decision-maker preferences. Moreover, a comparison with earlier approaches is carried out to evaluate the consis tency and accuracy of the proposed model’s ranking.
Intuitionistic fuzzy Z-numbers; weighted arithmetic averaging operator; weighted geometric averaging operator; MCDM.
Copyright (c) 2026 Zafar et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Entropy-Based Topological Characterization of Magnesium Silicide
Chemical graph theory focuses on the physical features of molecular structures using topological descriptors, where topological de scriptors are mathematical formulas that represents a molecular structure and can be applied to any graph. Chemical graph theory depends exten sively on topological descriptors, particularly in relation with chemical features of molecules. According to the structural characteristics of the graphs implemented in their computation, they can be categorised. This research paper focuses on several types of Zagreb descriptors for magne sium silicide’s molecular structure. The structure denoted by (Mg2Si), comprises of eight magnesium Mg and four silicon Si atoms. The struc ture and shape of the molecular graph are commonly described by the Zagreb indices. The significance of these indices in simulating the chemi cal characteristics of the magnesium silicide network is emphasised in the paper
Topological descriptors, Molecular complexity, Magnesium silicide, Graph entropy
Copyright (c) 2026 Ali Ahmad et al. https://creativecommons.org/licenses/by-nc-nd/4.0
Some New Results for Reverse Graph Energies and Graph Operations
In this paper, we establish the notions of reverse first Zagreb energy, reverse second Zagreb energy, and reverse degree square sum energy. We show how these energies behave across various graph families. Furthermore, we analyze the splitting graph and shadow graph energies, i.e. their context in the complete graph family. Our research adds to the overall knowledge about graph energies and their use within mathematical graph theory.
Saima Nazeer Department of Mathematics, Lahore College for Women University (LCWU), Lahore, Punjab, Pakistan
Copyright (c) 2025 Arooj Ibrahim, Saima Nazeer Department of Mathematics, Lahore College for Women University (LCWU), Lahore, Punjab, Pakistan, Sami Barhoumi Gafsa https://creativecommons.org/licenses/by-nc-nd/4.0
Optimal Bounds for the Mostar Index of Chemical Trees
Topological indices are the numerical values of a chemical graph which are uniquely defined for that graph. Topological indices play a pivotal role in Quantitative Structure-Property/Activity Relationship in-vestigations, offering a robust framework for elucidating intricate corre-lations between molecular architecture and physicochemical properties. They used to predict the bio-chemical activities of graphs. Topological in-dices constitute a specialized domain within chemical graph theory, hav-ing garnered significant attention in scholarly literature. Essentially, topo-logical indices provide a quantitative representation of molecular graphs, which can be visualized through various mathematical constructs, such as polynomials, numerical sequences, matrices, or singular values. Mostar index is one of the last distance based topological index. In this article, we discuss the Mostar index for chemical trees. Also we compute some upper bounds of chemical trees using Mostar index.
Zunera Shoukat Faculty of Sciences, The Superior University Lahore, Pakistan
Copyright (c) 2025 Hafiza Sana Sattar, Saima Mushtaq, Zunera Shoukat Faculty of Sciences, The Superior University Lahore, Pakistan https://creativecommons.org/licenses/by-nc-nd/4.0
Computational Modeling of Casson Blood Flow in Symmetrically Stenosed Carotid Arteries Based on Finite Volume Scheme
Carotid atherosclerosis is a major contributor to cardiovascular morbidity and mortality, primarily due to plaque-induced arterial steno-sis. This study develops a patient-specific computational model of Casson blood flow in symmetrically stenosed carotid arteries using a finite volume method. Realistic artery geometries are reconstructed from MRI and CT imaging. The model indicates a significant pressure drop (from approxi-mately 917 Pa to -411 Pa) and an increase in blood velocity, peaking at 1.5 m/s within the stenosed region. Temperature gradients are noted along ar-terial walls due to frictional heating. The model is quantitatively validated with reference data and clinically confirmed by MRI diagnostics. The main findings of computational results further show that wall shear stress increases with stenosis severity, and recirculation zones intensify near the stenosis throat, potentially promoting plaque buildup. The Casson fluid model captures the non-Newtonian nature of blood effectively, while the finite volume scheme ensures stable and accurate numerical predictions. These insights advance the understanding of hemodynamic behavior in stenosed arteries and may aid in personalized treatment planning.
Copyright (c) 2025 Abdul Wahab, Muhammad Imran Asjad , Kubra Anjum, Saraj Khan https://creativecommons.org/licenses/by-nc-nd/4.0
Architecting Resilient Communal Habitats: A Treatise on the Application of WheelGraphs to Security Optimization with Codes
Secure housing is a fundamental pillar of stable and thriving communities. It encompasses the provision of safe and reliable shelter, where individuals and families can live without fear of harm, theft, or intrusion. Security plan of a housing society can be constructed in many ways. In this article, authors have discovered a plan to strengthen security of a housing society by using face irregularity strength on wheel graphs which contains vertices, edges and faces. The vertices will act like houses, edges will be connections between houses and faces will represent the area among the houses. Mathematically, this research includes irregularity strength of face, vertex-face, edge-face and vertex-edge-face under Υ−labelling of class (σ, φ, ψ). An algorithm is established which will lead to the real life security of housing society.
Practical Applications of Irregularity Strength
Copyright (c) 2025 Abdul Aleem Mughal, Raja Noshad Jamil , Muhammad Reza Farahani https://creativecommons.org/licenses/by-nc-nd/4.0
Advanced Hemodynamic Modeling of Pulsatile Blood Flow with MagneticNano-particles: Fractional Analysis for Optimizing Biomedical Applications
Hemodynamic analysis plays a crucial role in the prevention, diagnosis, and treatment of human vascular diseases. The use of mag-netic particles in biomedicine and clinical therapies for targeted drug de-livery within cells provides simple diagnostic tools. In this article we will study the periodic pulsatile blood flow in the femoral and coronary ar-teries using nano-particles. Based on experimental data regarding blood rheology, the study employs the constitutive equation of an Oldroyd-B fluid. Moreover to capture more control, the problem is modeled using the ABC fractional derivative operator and solved through the application of integral transforms. Furthermore, a comprehensive graphical analysis is conducted to understand the influence of the fractional order parame-ter and various material factors, such as amplitude, lead angle, frequency of body acceleration, magnetic field, particle concentration, and tempera-ture effects, leading to significant conclusions. The obtained results help to optimize targeted drug delivery to infected tissues, and hyperthermia treatments.
Copyright (c) 2025 https://creativecommons.org/licenses/by-nc-nd/4.0
New Cryptosystem Using Sumudu Transform, Combination of Three Functions with Simulation and Python Code
In today’s world, information security is the most important component of our lives. Cryptography is among the best techniques designed to safeguard data security and message transmission. This paper presents a new method using successive combinations of the Sumudu transform of three functions for encoding and the inverse Sumudu transform for decoding. Starting with standard results on Sumudu transforms, we present our encryption method and obtain it as new theorems. Further, we generalized our method for the more secure algorithm. Finally, we used Python code to implement this strategy with simulation, our conclusions are based on the statistical analysis.
Copyright (c) 2024 Raut P. P., Hiwarekar A. P. https://creativecommons.org/licenses/by-nc-nd/4.0
A novel deep learning technique for solving dynamical model of infertile women with Polycystic Ovarian Syndrome Using the combination of radial basis and log-sigmoid functions
The focus of this study is to introduce the novel process of deep neural network (DNN) for the numerical solution of dynamical model based on infertile women with polycystic ovarian syndrome (PCOS). The model consists of five classes such as, Susceptible Women (S), Infertile Women (I), Women under Treatment 1 ( Women with Treatment 2 (and Recovered Women (R). The dual hidden layer neural network procedure is applied using the combination of radial basis function and sigmoid function for the numerical approximations of the nonlinear model. The twenty neurons for radial basis function and forty neurons for sigmoid function are used. The acquired numerical results are validated through the comparison of the solutions. The optimization is implemented by employing a competent scale conjugate gradient procedure. The verification of the proposed numerical method is observed through the reference results, and the reduced absolute error around is obtained. The reference database is achieved using the Runge-Kutta method of order 4. The method is proved reliable when trained, tested and validated using the reference dataset that consists of proportions 72%, 14% and 14%. After using the DNN approach, results are validated and displayed through MSE, regression, fitness, error and comparison plots.
Copyright (c) 2024 Ume Ayesha, Shumaila Javeed, Mansoor Shaukat Khan, Dumitru Baleanu, Mohsin Ali, Mustafa Baryam https://creativecommons.org/licenses/by-nc-nd/4.0
Bifurcation and Control of Rice Tungro Disease Spread in Plants Under Hypersensitive Response
The primary objective of this study is to investigate the spread of rice tungro disease within a community, with a particular focus on the role of predators in disease transmission. A mathematical model has been developed to examine the progression of rice tungro disease in a healthy environment, incorporating various control strategies such as the continuous removal of different categories of infected plants. To get proper numerical results, the rice tungro model is converted into a fractional rice model, and for fractional order, the fractal-fractional (FF) operator is used for continuous monitoring. The stability of the newly built model is checked by the quantitative and qualitative investigation. For a better understanding of the newly developed model, obtain the mathematical solution of the model with boundedness and uniqueness. The impact of different parameters on the spread of rice tungro disease is investigated. By using the Lipschitz condition and normed function, checking the spreads of rice tungro disease in all sub-compartmentss. Furthermore, the stability of the model is examined by the Hyers-Ulam concept. The flip bifurcation is investigated for all compartments at equilibrium points, and a simulation for flip bifurcation is examined and see the effect of different variables on the spreading of disease across all compartments for flip bifurcation. Additionally, for numerical analysis of the fractional operator used the two-step Lagrange polynomial method was used for the generalized form with Mittag-Leffler kernel. Finally, numerical simulations are employed to demonstrate the effects of various factors on disease dynamics. Simulations have been conducted to observe the actual behavior and progression of rice tungro disease at various stages, using different parameter values in a healthy environment exhibiting a hypersensitive response (HR). This research contributes to a deeper understanding of disease transmission and supports the development of effective management strategies for plants based on validated findings.
Copyright (c) 2024 Abdul Ghaffar, Khurram Faiz, Aqeel Ahmad, Ghulam Mustafa, Muhammad Farman https://creativecommons.org/licenses/by-nc-nd/4.0
Comparative Study of rth Chain Benzenoid Hex-Derived Network
This study integrates advanced algorithms with chemical graph theory to analyze the topological indices of benzenoid-derived nanostructures, focusing on their applications in computational chemistry and drug design. By developing novel mathematical formulations for degree-based indices (Rα, M1, H, AZI, ABC, and GA), we establish quantitative relationships between structural parameters (r, s) and physicochemical properties of hexagonal networks. Our results reveal that increasing network dimensions enhances molecular stability and electron delocalization in these nanostructures, offering critical insights for optimizing antiviral agents and energy storage materials. The proposed computational framework, validated through rigorous graphical and tabular comparisons, provides a robust tool for predicting structure-activity relationships in drug discovery and designing next-generation nanomaterials. This work bridges theoretical graph theory with practical applications in nanotechnology and pharmaceutical sciences, demonstrating significant potential for sustainable innovation in medical and energy technologies.
Copyright (c) 2024 Haidar Ali, Rimsha Zahid, Muhammad Asif, Barya Iftikhar, Parvez Ali https://creativecommons.org/licenses/by-nc-nd/4.0
Domain Wall’s Impact on a Type III Bianchi Universe with f(R, T) Gravity
In this research we employ first model of f(R,T) gravitational theory f(R, T) = R + 2f(T) to explore the type III Bianchi Universe featuring domain wall. It was assumed that the expansion scalar is directly related to the shear scalar to find solution of the field equation. This results the metric potentials connection Z = kY n and the consideration of two scenarios for the integrating constant D. We observed that the domain wall existed early in the universe’s history and eventually
Copyright (c) 2024 M. Junaid, N. P. Gaikwad https://creativecommons.org/licenses/by-nc-nd/4.0
Some Remarks on Results Related to Ostrowski−Gruss Inequality ¨
The article [14] published in 2007 titled “New bounds for the first inequality of Ostrowski−Gruss type and applications in numerical in- ¨ tegration” contains very important results in the sense that the main idea of the article is unique, and it helps to generalize many established results. In the article [11], which appeared in the year 2013, we find exactly the same proving techniques used, leading to similar but slightly different results as compared to results stated in [14]. This arises the question: Why do two different authors, using exactly same proving techniques, arrive at different results? Now we compare results of the two articles and give some remarks on it and produce a better version.
Copyright (c) 2024 Asif R. Khan https://creativecommons.org/licenses/by-nc-nd/4.0
Existence and Uniqueness of Solutions for Atangana-Baleanu Fractional Differential Equations in Caputo-Sense
In this work, we investigate the existence and uniqueness of Atangana-Baleanu solutions in Caputo sense fractional differential equations. The existence results are based on Monch’s fixed point theorem. ¨ The idea of uniqueness is investigated using the Banach contraction principle. Furthermore, a practical example of the results of solutions that include various fractional orders is provided to validate the theoretical insights provided in this paper.
Copyright (c) 2024 Abduljawad Khairi Anwar https://creativecommons.org/licenses/by-nc-nd/4.0
A Generalized Fractional Integral Transform for Solutions of Fractional Burger’s Equation
This paper introduces a novel fractional-order integral transform within the field of fractional calculus and applies it to the solution of fractional burger’s equation with different fractional differential operator. In this study, we apply the newly proposed transform to several fractional differential, including Caputo, Caputo-Fabrizio, RiemannLiouville, New Fractional Derivative and Atangana-Baleanu operators in both the Riemann-Liouville and Caputo senses. For varying values of ϕ α(s), ψ(s) and γ(t), the over 200 existing integral transforms and fractional integral transforms can be considered special cases of the proposed transform when applied to the aforementioned derivatives. This suggests the versatility and applicability of our newly introduced fractional-order integral transform within the broader context of fractional calculus, engineering and physics. The analytical solution of Fractional order Viscous Burger’s equation with different differential and integral operators are also discussed.
Copyright (c) 2024 Sidra Younis, Noreen Saba, Ghulam Mustafa, Muhammad Asghar, Faheem Khan https://creativecommons.org/licenses/by-nc-nd/4.0
Fibonacci numbers which are concatenation of three Fibonacci or Lucas numbers
In this note, we show that there is no Fibonacci number that can be expressed as a concatenation of three Fibonacci or Lucas numbers under a certain constraint. That is, we solve to the Diophantine equations Fn = 10d+lFm1 + 10lFm2 + Fm3 and Fn = 10d+lLm1 + 10lLm2 + Lm3 in non-negative integers (n, m1, m2, m3) with m2 ≤ m1, where d and l represent the number of digits of the (Fm2, Lm2) and (Fm3, Lm3), respectively. Moreover, we give these equations as a problem to the researchers without the constraint m2 ≤ m1.
Copyright (c) 2024 Fatih Erduvan https://creativecommons.org/licenses/by-nc-nd/4.0
Partial Differential Equations Possessing New Complex Derivatives
The recently proposed new operator for complex differentiation is applied to partial differential equations for the first time. The definition involves complex numbers in the differentiation operators. In this sense, it does not possess any resemblance with the real number differentiation of complex valued functions. First, the operator is defined with its basic properties. Several partial differential equations are considered involving the new complex number derivatives. First, the homogenous constant real coefficient linear partial differential equations are considered. If the real and imaginary components of the derivatives are equal, the imaginary parts can be ignored. For complex valued coefficients, for non-homogenous equations and for derivatives having different real and imaginary parts, the imaginary parts cannot be ignored. Some nonlinear partial differential equations are also considered. In the calculations, the Cauchy-Riemann as well as the Laplace equations appears frequently. Expression of the Hamilton’s equations and the Schrodinger equation in ¨ terms of the new differential operators are outlined. The new operators have the potential to be applied to mathematical modeling of several physical problems in the future
Nonlinear Partial Differential Equations
Copyright (c) 2024 Mehmet Pakdemirli https://creativecommons.org/licenses/by-nc-nd/4.0