VFAST Transactions on Mathematics
type: channel
description: The VFAST Transactions on Mathematics is a biannually peer-reviewed scientific journal published by the VFAST-Research Platform. It was established in 2013 and covers the areas of Pure and Applied Mathematics and its applications. The goal of this journal is to provide an international platform to Mathematicians all over the world to promote, share, and discuss various new issues and development in the field of Mathematics and its application. ISSN: 2309-0022 (Online), 2411-6343 (Print) Editor in Chief: Dr. Muhammad Arif, Associate Professor, Abdul Wali Khan University Mardan <span class="csspropertycolor" style="box-sizing: inherit; color: red; font-family: Consolas, Menlo, 'courier new', monospace; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-decoration-thickness: initial; text-decoration-style: initial; text-decoration-color: initial;">Important Links REGISTER SUBMISSION EDITORIAL TEAM LATEX TEMPLATE POLICY DOCUMENT The Journal is indexed by: HEC Recognized in Category Y Google Scholar Journal Tocs Repindex Research Gate World of Periodicals University of Illinois Library Ulrich’s Periodicals Directory/ProQuest <a href="http://sfx.princeton.edu:9003/sfx_pul/az?param_letter_group_save=&param_chinese_checkbox_type_save=Pinyin&param_current_view_save=table&param_langcode_save=en&param_pattern_save=VAWKUM+Transaction+on+Computer+Sciences&param_services2filter_save=getFullTxt&param_jumpToPage_save=&param_chinese_checkbox_save=0&param_starts_with_browse_save=0&param_sid_save=b2d32284a5f202b8c792b440b6535c2f&param_type_save=textSearch&param_lang_save=eng&param_textSearchType_save=startsWith&param_perform_save=searchTitle&param_letter_group_script_save=&param_type_value=textSearch&param_jumpToPage…
publisher: VFAST-Research Platform
language: en-US
publication name: VFAST Transactions on Mathematics
issn: 2411-6343
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items: A Comparative Study on Predictive Modeling of T20 Opener Success in International Cricket Tournaments via Machine Learning Models
items: Improved exponential ratio-cum-regression type estimators under stratified random sampling for population mean
items: An Extended TOPSIS Method for the Multiple Attribute Decision Making Problems Based on double hierarchy linguistic soft sets (DHLSSs)
items: Double Hierarchy Linguistic Soft and Soft Covering Based Sets and Their Application
items: Investigating Reactive Species Diffusion in Power-Law Fluids Beyond Stretched Surfaces
items: Solutions of Real-World Problems Using Transportation Method
items: New Numerical Insights into Electromagnetic Mass in Spherically Symmetric Configurations
items: Impact of Individual Participation and Their Health Education on the Growth of the Sportsy Population
items: Establishing Neural Network Models for Predicting Flood Propagation and Recession in Urban Roads
items: Predictive Modeling of Parameter Variations in Conveyor Belt Dryers using Machine Learning for Improved Drying Performance
items: A Novel Hesitant Cubical Dombi Fuzzy Aggregation Operators for Selecting Green Supplier Chain Managements
items: Improving Efficiency of Finite Population Distribution Function Using Twofold Auxiliary Information Based on Simple Random Sampling
items: A study of coefficient-related problems for symmetric starlike functions connected with a tan hyperbolic function
items: Improved Estimator for the Estimation of Population Mean Using a Predictive Approach Under PPS Sampling
items: Efficient Estimation of Population Variance Using a Novel Optional Scrambling Model
items: Several Topological Indices and Entropies for Certain Families of Commutative Graphs over Quaternion Groups
items: Comparative Analysis of Time Series Forecasting using ARIMA, and GRNNs Models: A Case Study of Death Rate of Diabetic Mellitus in Canada
items: Numerical Simulation Model of the Infectious Diseases by Comparing Backward Euler Method and Adams-Bash forth 2-Step Method
items: Geometric Modelling of a Family of 3-point Quaternary Subdivision Schemes Rζ
items: Parallel Numerical Solution of 2D Electrostatics Poisson Equation on Different Mesh Partitioning Schemes
items: Time Series Modeling and Forecasting of the Patients’Inflow and Admission in the Hospitals: A cases study of LUMHS Hospital Jamshoro Pakistan
items: Nonparametric test for multiple crossing Survival Curves
items: Improved Class of Regression Type Mean Estimators in Simple Random Sampling Using Concomitant Variable: An Application in T-20 International Cricket
items: Mathematical and computationalmodeling of migrations, population spread, and Location-dependent carrying capacities
items: Efficient Class of Variance Estimators for Population using Supplementary Information in Stratified Random Sampling
items: Geometric Modelling of a Family of 4-Point Ternary Approximating Subdivision Scheme U_φ with Visual Performance
items: Investigation the Stochastic behaviour of the Traffic Flow: A Case Study of a Section of a Road
items: Improved Mean Estimators for Population utilizing Dual Supplementary Characteristics under Simple Random Sampling
items: A Study of Mathematical Models Used in Anaerobic Digestion of Organic Refuse
items: Homotopy Perturbation Method with Analytics for solving Bivariate type II Fuzzy Fredholm Integral Equations
items: An Improved Blended Numerical Root-Solver for Nonlinear Equations
items: Series Solution of Unsteady Tank Drainage for Third Order Fluid using Adomian Decomposition Method
items: Modiﬁed Bracketing Iterative Method for Solving Nonlinear Equations
items: Order Structured Graphs of Cyclic Groups and their Classification
items: Patterns of Nanoporous Spherical Packing Emerging under Influence of Curvature and Confinement
items: A versatile optional randomized response technique for use with sensitive surveys
items: New Results for Riemann Solution of the Cargo-LeRoux Model by the Application of Flux-Limiter Schemes
items: Quadratic twist of an elliptic curve in a generalized Weierstrass equation over a function field
items: Exact Solution of Tank Drainage for Bingham Fluid flow circular tube
items: Some New Generators to Produce Minimal Circular Strongly Balanced Generalized Neighbor Designs of Class-I

A Comparative Study on Predictive Modeling of T20 Opener Success in International Cricket Tournaments via Machine Learning Models
type: item
description: In the game of cricket, particularly during high-stakes tournaments, players’ performances have substantial consequences for their teams and energetic crowd. Predicting players’ outcomes is often validated by experts’ territory through mathematical and statistical models. However, due to the intricacies of cricket, player-related features in different sports cannot be evaluated comparatively. De spite these challenges, the rising utilization of Machine Learning (ML) models has proven crucial role in achieving precise predictions. In this research study, the ultimate aim was to predict the performance of T20 opening batters for upcoming T20 tournaments. Player records were compiled from ESPNcricinfo and Cricbuzz. Several ML models are implemented to predict players’ outcomes. The analysis for this study was categorized into two cases: runs scored and strike rate, acknowledging both pre-match and all-match features. For predicting outcomes based on runs scored using pre-match features, Decision Tree and Naïve Bayes outperformed with an accuracy of 0.75, while for strike rate, K-Nearest Neighbor surpassed models with an accuracy of 0.68. Furthermore, assessing players’ performance on runs scored using all-match features, Naïve Bayes and Support Vector Machine achieved exceptional accuracy of 0.98. For strike rate across all-match features, logistic regression beat the models with a leading accuracy of 0.98.
creator: Syed Muhammad Arsalan
creator: Syed Muhammad Zeeshan
rights: Copyright (c) 2026 VFAST Transactions on Mathematics
date: 2025-12-31
publication date: 2025-12-31
volume: 13
number: 2
starting page: 55
ending page: 74
doi: 10.21015/vtm.v13i2.2207

Improved exponential ratio-cum-regression type estimators under stratified random sampling for population mean
type: item
description: This research introduces a distinctive class of exponential ratio-cum-regression estimators under the technique of stratified random sampling (STRS) , designed for the efficient assessment of the population mean by incorporating dual concomitant variables (CV). Six refined estimators are introduced, by deriving their mean square error (MSE) through the approximation of first-order. These derivations are carried out through Taylor expansion technique. Their performance is systematically assessed against existing alternatives based on the MSE criterion. Theoretical developments supported by empirical evidence, and reveal that the proposed estimators consistently offer reduced MSE and enhanced percentage relative efficiency (PRE) over conventional methods.
creator: Shumaila Wazir
creator: Muhammd Atif
creator: Mujeeb Hussain
rights: Copyright (c) 2026 VFAST Transactions on Mathematics
date: 2025-12-31
publication date: 2025-12-31
volume: 13
number: 2
starting page: 75
ending page: 89
doi: 10.21015/vtm.v13i2.2054

An Extended TOPSIS Method for the Multiple Attribute Decision Making Problems Based on double hierarchy linguistic soft sets (DHLSSs)
type: item
description: In this paper, classical soft set theory is extended to develop Double Hierarchy Linguistic Soft Sets (DHLSSs) for addressing multi-attribute decision-making problems. DHLSSs provide an effective framework for handling qualitative information expressed through double hierarchy linguistic terms. Since the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a widely accepted method for multi-criteria decision analysis, this study integrates TOPSIS with DHLSSs to propose an extended decision-making approach. The proposed method is designed to handle situations in which attribute weights and attribute values are unknown and represented in the form of double hierarchy linguistic term elements. Background concepts related to DHLTSs, Hamacher t-norms and t-conorms, as well as fuzzy sets, rough sets, soft sets, and fuzzy soft sets are briefly reviewed. Attribute weights are determined using an entropy-based method, and an improved TOPSIS algorithm is employed to rank alternatives. An illustrative example is presented to demonstrate the feasibility and effectiveness of the proposed approach.
creator: Muhammad Javed Khan
creator: Hamza Zafar
rights: Copyright (c) 2026 VFAST Transactions on Mathematics
date: 2025-12-31
publication date: 2025-12-31
volume: 13
number: 2
starting page: 41
ending page: 54
doi: 10.21015/vtm.v13i2.2161

Double Hierarchy Linguistic Soft and Soft Covering Based Sets and Their Application
type: item
description: Recent developments in mathematics have resulted in the creation of the soft set theory as a method for dealing with uncertainty. The classical soft sets, however, are not well suitable for dealing with double hierarchy linguistic term parameters. This study's objective is to expand traditional soft sets to include double hierarchy linguistic soft sets and double hierarchy linguistic soft covering based sets, whereby the DHLTSs, soft sets, and soft covering-based sets merge. Next, we describe upper and lower approximation, positive, negative, and border regions for double hierarchy linguistic soft sets and double hierarchy linguistic soft covering based sets. Ultimately, the double hierarchy linguistic soft set is applied to a decision-making issue with the help of fuzzy soft sets, and the value is demonstrated with a numerical example. In this example, we chose the best suitable Renewable Energy Source (RES) alternative for installation at Abdul Wali Khan University Mardan (AWKUM), using the recently established MCDM approach VIKOR method.
creator: Javed Khan
creator: Hamza Zafar
creator: Marya Nawaz
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2025-06-30
publication date: 2025-06-30
volume: 13
number: 2
starting page: 41
ending page: 58
doi: 10.21015/vtm.v13i1.2155

Investigating Reactive Species Diffusion in Power-Law Fluids Beyond Stretched Surfaces
type: item
description: In this study, a numerical solution is developed for the steady flow of a power-law fluid influenced by magnetohydrodynamics (MHD) effects over a continuously moving, linearly stretching surface. The model incorporates mass transfer involving species concentration and a one-stage chemical reaction under isothermal and homogeneous conditions. Reactive species emitted from the surface diffuse into the fluid, driven solely by the sheet's motion.The model examines the effects of six key parameters: power-law index, magnetic parameter, modified Schmidt number, reaction rate parameter, wall concentration parameter, and local Grashof number. These factors significantly influence the mass transfer characteristics and fluid flow behavior.To solve the governing equations, an artificial neural network (ANN) combined with the Levenberg-Marquardt Algorithm (LMA) is employed. The differential equation is reformulated into an ANN-based optimization problem, minimizing the mean square error as a fitness function. The ANN-LMA results are compared graphically and numerically with reference solutions for validation.Graphical results show that the magnetic field increases surface skin friction but slightly decreases the mass transfer rate. However, the mass transfer is highly sensitive to changes in the modified Schmidt number and the reaction rate parameter, increasing with their higher values. Statistical measures such as mean, standard deviation, minimum, and maximum values are used to further validate the solution.Overall, the study highlights the importance of exploring non-Newtonian fluid models, especially those exhibiting shear-thinning behavior, which appears to lower wall shear stress and adds complexity to power-law fluid dynamics.
creator: Muhammad Gul Zeb Khan
creator: Muhammad Fawad Khan
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2025-06-30
publication date: 2025-06-30
volume: 13
number: 2
starting page: 01
ending page: 18
doi: 10.21015/vtm.v13i1.2117

Solutions of Real-World Problems Using Transportation Method
type: item
description: The paper is an analysis of the transportation problem which is one of the most basic and popular optimization problems in operational research. The transportation problem is aimed at finding the most cost-efficient means of distributing a homogenous product, usually a factory or origin, to various demand points, usually a warehouse or destination. Every source is characterized by the fixed amount of supply, and every destination is characterized by the particular demand that needs to be addressed. The main aim of the transportation model is to reduce the overall transportation cost and at the same time make sure that all the supply and demand restrictions are met. It is assumed that the means of transporting a unit of product to a specific warehouse is known and constant. The decision variables are the amount of goods carried within each route generated by variations of the origins and destinations. Through the proper choice of these variables, the model determines the most optimum plan of transportation that will give the lowest overall cost. This paper is the formulation of the mathematical model of the transportation problem, its objective function, and the constraints associated with it. The objective function can be established as the result of multiplication of the unit transportation costs and the decision variables of all routes. The constraints are designed to make sure that the overall number of goods shipped by each factory will not be beyond what it can supply and also the overall amount that is received by each warehouse will satisfy its needs. There are also non-negativity restrictions that are applied to ensure that all amounts of shipments are practical. To understand it better, the simplistic model of the transportation issue is depicted through the tabular representation, in which the costs, supplies, demands, and allocation variables are evident. The constraints and decision variables interaction is discussed to show how feasible and optimality s…
creator: Shakoor Muhammad
creator: Khalil Ullah
creator: Waqar Ahamd
creator: Syed Muhammad Ghufran
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2025-06-30
publication date: 2025-06-30
volume: 13
number: 2
starting page: 84
ending page: 106
doi: 10.21015/vtm.v13i1.2109

New Numerical Insights into Electromagnetic Mass in Spherically Symmetric Configurations
type: item
description: Our study aims to evolve artificial intelligence that emerges from natural processes and be able to predict nonlinearities in the lane Emden-Fowler (LEF) equation. More precisely, the feedforward artificial neural network model is an adaptive one that leads to accurate solutions of the LEF equation. This dataset concerns a neural network whose parameters have been made adjustable so that initial predictions can be based on a given model quite easily. The energy reduction objective function for the specific limitations of the LEF equations is based upon contextual nuances introduced in previous optimization process. To begin with, this proposed methodology has been tested through experiments and initial conditions affecting the initial value of any such problem for LEF. We consider three cases in detail which show how our method can solve the LEF equation effectively. Our combined method (PSO-GWO-IPA) with Particle Swarm Optimizer (PSO) and Grey Wolf Optimizer (GWO) achieves very good convergence speed when compared to PS, IPA, PSO, PS-IPA, HPM and OHM. Through statistical tests, we will verify reliability and validity of our approach mentioned above. Our empirical results are in perfect agreement with the mathematical model, demonstrating the wisdom of the proposed method.
creator: Tauqeer Ahmad
creator: Fazl Ullah Fazal
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2025-06-30
publication date: 2025-06-30
volume: 13
number: 2
starting page: 19
ending page: 40
doi: 10.21015/vtm.v13i1.2108

Impact of Individual Participation and Their Health Education on the Growth of the Sportsy Population
type: item
description: Health education plays a significant role in promoting health, creating awareness of health issues, and adjusting the focus on the value of physical exercise in enhancing the general health of individuals. This paper focuses on a mathematical model for the interaction between health education, individual involvement, and the growth of the sports population and proposes the best decision solutions. An intelligent approach based on Bayesian Regularization (BR) and neural networks is used to predict the solutions of the presented mathematical model. To verify the accuracy of the predicted solutions for the mathematical model, the RK-4 numerical technique is utilized to generate target points for comparison. Different scenarios and the subsequent variation in parameters are used to determine the solutions of the model and the stability of our utilized technique. This study provides useful information for developing effective evidence-based interventions to promote a healthier and more active population.
creator: Hamad Jan
creator: Muhammad Sulaiman
creator: Muhammad Saad
creator: Awais Khan
creator: Muhammad Salman
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2025-06-30
publication date: 2025-06-30
volume: 13
number: 2
starting page: 59
ending page: 83
doi: 10.21015/vtm.v13i1.2107

Establishing Neural Network Models for Predicting Flood Propagation and Recession in Urban Roads
type: item
description: In this analysis, a contagion model is a straightforward yet effective mathematical approach—that was used to forecast the temporal change of the outset and contiguous distribution and recession of flooding in metropolitan roadway networks. A system of metropolitan roadways must be flood-resistant in order to provide public services and deal with emergencies. The dispersion of floodwaters is a complicated temporal-spatial process that affects urban networks. In comparison to the SEIR (Susceptible-Exposed-Infected-Recovered) prototype, a system of ordinary differential equations, four macroscopic characteristics, rate of including flood spreading represented by ($\beta$), rate of flood incubation symbolized by ($\alpha$), and rate of recovery highlighted by ($\mu$), can be used to understand how floods evolve within networks. Additionally, by joining the backpropagated neural network and the Levenberg—Marquardt algorithm (NN-BLMA), surrogate solutions to the model are discovered. This method has some clear advantages over conventional ones, including flexibility, comparatively simple implementation, and fastest results. Reference solutions are generated using the Runge-Kutta of order four (RK4) method. We have examined three distinct scenarios to analyze our surrogate solution models. By changing $\alpha$, $\beta$, $\mu$, and k, the mathematical model's stability and equilibrium are examined. To gauge the validity of our machine learning process, we categorize our candidate solutions into training, testing, and experimental class. The efficacy of the NN-BLMA scheme has been confirmed by comparative examinations of statistical values based on mean squared error function (MSEF), effectiveness, regression plots, and failure histograms.
creator: Muhammad Fawad Khan
creator: Fazl Ullah Fazal
creator: Awais khan
creator: Muhammad Saad
rights: Copyright (c) 2026 VFAST Transactions on Mathematics
date: 2025-12-31
publication date: 2025-12-31
volume: 13
number: 2
starting page: 20
ending page: 40
doi: 10.21015/vtm.v13i2.2106

Predictive Modeling of Parameter Variations in Conveyor Belt Dryers using Machine Learning for Improved Drying Performance
type: item
description: This study takes an in-depth look at the mathematical model governing the operation of a tangential flow conveyor dryer operating in a co-current configuration. The Conveyor Belt Dryer (CBD) is represented as an Ordinary Differential Equation (ODE), and our research focuses on studying the influence of parameter variation and symmetry on the Rate of Exchange of Moisture Content (RMC). To achieve this, we employ a system studying framework based on artificial Neural Networks (NN) and the Levenberg-Marquardt Training (LMT) algorithm, enabling the examination of symmetry within surrogate solutions. The RK-four method is applied to generate target data points for supervised learning in the NN-LMT structure. Furthermore, we investigate various scenarios of the mathematical model related to the rate of change of moisture content. Detailed graphical representations, including histograms, absolute error plots, curve fitting graphs, and regression graphs, are employed to facilitate comprehensive explanations. Additionally, a comparative analysis between the numerical solutions obtained through the machine learning technique is provided, followed by graphical and statistical representations of the determined errors.
creator: Awais Khan
creator: Tauqeer Ahmad
creator: Naveed Ahmed Khan
creator: Muhammad Sulaiman
rights: Copyright (c) 2026 VFAST Transactions on Mathematics
date: 2025-12-31
publication date: 2025-12-31
volume: 13
number: 2
starting page: 01
ending page: 19
doi: 10.21015/vtm.v13i2.2105

A Novel Hesitant Cubical Dombi Fuzzy Aggregation Operators for Selecting Green Supplier Chain Managements
type: item
description: A hesitant fuzzy (HF) set enhances the concept of fuzzy sets by addressing disagreements among decision-makers about the membership degree of an element. Similarly, the Cubical Fuzzy Set (CFS) is useful for managing uncertainty in decision-making problems. However, existing methods often lack integration of hesitation and cubical uncertainty, and there is limited exploration of their combined effects on aggregation processes. In this paper, we introduce the Hesitant Cubical Fuzzy Set (HCFS), which integrates the principles of HF sets and CFS to address these limitations. We define several set-theoretical operations for HCFSs and develop Dombi operations for them. Furthermore, we present a range of aggregation operators based on Dombi operations, including the Hesitant Cubical Dombi Fuzzy Weighted Arithmetic Averaging (HCDFWAA) Operator, the Hesitant Cubical Dombi Fuzzy Weighted Geometric Averaging (HCDFWGA) Operator, the Hesitant Cubical Dombi Fuzzy Ordered Weighted Arithmetic Averaging (HCDFOWAA) Operator, and the Hesitant Cubical Dombi Fuzzy Ordered Weighted Geometric Averaging (HCDFOWGA) Operator, and examine their properties. Additionally, we propose a multi-criteria group decision-making method and algorithm within the Hesitant Cubical Fuzzy framework. To address gaps in practical application, we provide an example of the selection of green suppliers in supply chain management. We also perform a comparative analysis with existing operators to highlight the advantages and effectiveness of our approach, emphasizing how the integration of hesitation and cubical uncertainty can enhance decision-making processes.
creator: Asghar Khan
creator: Tahir Zaman
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2024-09-30
publication date: 2024-09-30
volume: 13
number: 2
starting page: 67
ending page: 98
doi: 10.21015/vtm.v12i2.2006

Improving Efficiency of Finite Population Distribution Function Using Twofold Auxiliary Information Based on Simple Random Sampling
type: item
description: The main purpose of this article has been suggested two modified ratio type estimators for estimation of finite population distribution function using auxiliary information under simple random sampling. Mathematical expressions such as Mean Square Error (MSE) and Bias are examined up to the first order approximation for all considered estimators in this article. For the ideal value of the kappa constant (K), the minimum MSE value for the recommended estimators has been found. Five actual data sets have been carried out to check the precision of the suggested estimators. The suggested estimators found to be superior and more efficient than the existing estimators for population distribution function. The theoretical and empirical comparisons are also conducted. Hypothetically, the proposed estimators perform better than the existing estimators.
creator: Muhammad Junaid
creator: Sadaf Manzoor
creator: Amjid Ali
creator: Manahil SidAhmed Mustafa
creator: Sohaib Ahmad
creator: Sardar Hussain
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2024-09-30
publication date: 2024-09-30
volume: 13
number: 2
starting page: 49
ending page: 66
doi: 10.21015/vtm.v12i2.1983

A study of coefficient-related problems for symmetric starlike functions connected with a tan hyperbolic function
type: item
description: The article aims to determine the sharp bounds of coefficients, Fekete-Szegö, Zalcman inequalities for the family SS∗ tanh of starlike function with respect to symmetric points linked with tan hyperbolic function. We also estimate determinant of |H2,2 (f)| is also obtained for the same class. Further, we study the logarithmic and inverse coefficients for the same class.
creator: Siraj Osman Omer
creator: Muhammad Aamir
creator: Muhammad Bilal
creator: Khalil Ullah
creator: Abbas Qadir
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2024-12-31
publication date: 2024-12-31
volume: 13
number: 2
starting page: 99
ending page: 113
doi: 10.21015/vtm.v12i2.1945

Improved Estimator for the Estimation of Population Mean Using a Predictive Approach Under PPS Sampling
type: item
description: In this study, we apply a predictive approach to the problem of finding new estimators for the estimation of finite population mean using auxiliary variable under probability proportional to size (PPS) sampling. This work is deemed novel due to the fact that, as far as we are aware, no one has before investigated the predictive approach to estimating the finite population mean under probability proportional to size sampling. The expressions for the bias and mean square error (MSE) are derived up to the first order. In order to verify the theoretical results, numerical and simulation investigations are conducted respectively. Based on the numerical result, it is shown that the suggested estimator performs well in terms of minimum MSE and higher percentage relative efficiency (PRE). The conditions under which the suggested estimator is more efficient than the other estimators are described numerically.
creator: Sajid Khan
creator: Muhammad Farooq
creator: Sohaib Ahmad
creator: Sardar Khan
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-09-30
publication date: 2024-09-30
volume: 13
number: 2
starting page: 01
ending page: 16
doi: 10.21015/vtm.v12i2.1942

Efficient Estimation of Population Variance Using a Novel Optional Scrambling Model
type: item
description: In the past few years, survey researchers have developed variance estimators of sensitive variables under randomized response techniques. \textcolor{black}{ The available variance estimators utilize linear scrambling models in which all of the survey participants are forced to scramble their responses and thus hide their true responses. In practice, some of the respondents may have no problem in reporting their true response to the researcher. The current study finds that using a true response option produces more efficient estimates of population variance compared to a linear scrambling model.} Additionally, we also suggest a new variance estimator of a sensitive variable of interest and analyze its algebraic properties using an auxiliary variable. We also conduct a simulation study to show the improvement over the existing estimators of the population variance.
creator: Muhammad Azeem
creator: Abdul Salam
creator: Ammara Cheema
creator: Abdul Basit
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2024-09-30
publication date: 2024-09-30
volume: 13
number: 2
starting page: 17
ending page: 31
doi: 10.21015/vtm.v12i2.1913

Several Topological Indices and Entropies for Certain Families of Commutative Graphs over Quaternion Groups
type: item
description: A group graph is a type of graph formed by combining a group, usually a finite group, with a generating set for that group. Group graphs are employed in various mathematical situations, including algebraic and computational group theory. A graph G is known as a commutative graph if the vertex set of G is a group and two elements are adjacent to each other if they are commuting to each other. In this work, we consider the family of commutative graphs over Quaternion groups. The edge partition mappings related to the degree of each vertex of the graph G are computed. Further, we established many results on various kinds of topological indices and entropies by using M-polynomials. The numerical comparison among computed topological indices has been proposed.
creator: Shahbaz Ali
creator: Muhammad Khalid Mahmood
creator: Sobia Ghaffar
rights: Copyright (c) 2025 VFAST Transactions on Mathematics
date: 2024-09-30
publication date: 2024-09-30
volume: 13
number: 2
starting page: 32
ending page: 48
doi: 10.21015/vtm.v12i2.1901

Comparative Analysis of Time Series Forecasting using ARIMA, and GRNNs Models: A Case Study of Death Rate of Diabetic Mellitus in Canada
type: item
description: This research aims to compare ARIMA and GRNN models alone. For this comparison the death rate for diabetes mellitus time series data of Canada is used. Autoregressive Integrated Moving Average (ARIMA), and Generalized Regression Neural Networks (GRNN) models were applied for time series prediction of the death rate for diabetes mellitus—trained data for two models from 2000 to 2015. Test data was used to compare the precision through data from 2016 to 2021. The ARIMA model was applied using the auto-command through R package which provided the least BIC and AIC values. The mean absolute deviation (MAD), root mean squared error (RMSE), and mean absolute percentage error (MAPE) were employed to measure the forecasting efficiency of the models. The ARIMA model had the highest prediction accuracy as compared to the GRNN model. ARIMA predicts the death rate for diabetes mellitus more accurately and robustly compared to the GRNNs model.
creator: Muhammad Shahbaz Khan
creator: Mir Ghulam Hyder Talpur
creator: Muhammad Aslam
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 415
ending page: 423
doi: 10.21015/vtm.v12i1.1894

Numerical Simulation Model of the Infectious Diseases by Comparing Backward Euler Method and Adams-Bash forth 2-Step Method
type: item
description: In this work, the Backward Euler technique and the Adams-Bashforth 2-step method—two numerical approaches for solving the SIR model of epidemiology are compared for performance. An essential resource for comprehending the transmission of infectious illnesses like COVID-19 in the SIR model. While the explicit Adams-Bash forth 2-step approach is well known for its computing efficiency, the implicit Backward Euler method is noted for its stability. The study evaluates the accuracy, strength, and computing cost of the two approaches to determine which approach is best for simulating the spread of infectious illnesses. The SIR Model was easily solved using the Adams Bashforth 2-step analysis and the Backward Euler method. The approaches' solutions are close to the exact requirements. There are important distinctions between the two-step Adams Bashforth and backward Euler procedures. The running time of the Adams Bashforth 2-step backward Euler method is shorter than that of the backward Euler method.
creator: Paras Soomro
creator: Israr Ahmed
creator: Faraz Ahmed Soomro
creator: Darshan Mal
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 402
ending page: 414
doi: 10.21015/vtm.v12i1.1881

Geometric Modelling of a Family of 3-point Quaternary Subdivision Schemes Rζ
type: item
description: Computer-aided geometric design combines mathematical concepts and computing skills that smooth curves through subdivision schemes. Subdivision schemes perform smoothing by turning the control polygon into a limit curve under a refinement rule, a prime example of which is improving the signal-to-noise ratio in modern devices. Because of the importance and location of subdivision schemes, mathematicians use them in CAD, computer graphics and advanced simulation methods. In this research, a family of 3-point Quaternary approximating subdivision scheme $R_{\zeta}$ is presented with its properties and analysis, including necessary conditions for convergence, Laurent polynomial, degree of the generation and polynomial reproduction, continuity analysis, Hölder regularity, and limit stencils. The visual performance of the proposed scheme is also presented to highlight the importance of this research and to validate the scheme.
creator: Rabia Younis
creator: Muhammad Javed Iqbal
creator: Inayatullah Soomro
creator: Usama Gulzar
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 379
ending page: 401
doi: 10.21015/vtm.v12i1.1868

Parallel Numerical Solution of 2D Electrostatics Poisson Equation on Different Mesh Partitioning Schemes
type: item
description: The ideas of parallelism for the large scale problems or problems with dense meshes have gained much attention in last few decades. The key goal of applying the parallelization is to reduce the computational time. In this paper; the 2D finite difference mesh partitioning schemes and their effect on performance of parallel numerical solution is evaluated. The main objective was to investigate the mesh partitioning schemes for less computational time and high speedup. For testing and implementation purpose a 2D electrostatics Poisson’s equation with Dirichlet and Neumann boundary conditions applied on a 2D cross section of Electrohydrodynamic (EHD) planar ion-drag micropump is used to simulate the electric potential and electric field on a parallel system. The performance of the 7 different mesh partitioning schemes (PS) in terms of computational time, speedup, efficiency and communication cost was evaluated. It was revealed that among the seven different partitioning schemes the PS-3 (two-way or tile partitioning) is found the best scheme for the parallel numerical simulation of the problem. Moreover, the parallel algorithm remains more efficient on \(P=2\) to \(P=8 \) workers while for \(P>8\) the efficiency of the algorithm may drop because of the high communication time.
creator: Shakeel Ahmed Kamboh
creator: Faiza Khanam
creator: Nadeem Naeem
creator: Sajida Parveen
creator: Sakina Kamboh
creator: Safina Kamboh
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 323
ending page: 335
doi: 10.21015/vtm.v12i1.1847

Time Series Modeling and Forecasting of the Patients’Inflow and Admission in the Hospitals: A cases study of LUMHS Hospital Jamshoro Pakistan
type: item
description: The patients’ crowding in the hospitals is an international phenomenon that demands much attention to avoid harm to the lives of patients. The quantitative based models have been successfully investigated to predict the crowding of patients. Thus, the main objective of this study is to probe a statistically feasible forecasting model capable of estimating the crowding of patients (patients’ inflow and patients’ admission specifically). As a case study, the Liaquat University of Medical and Health Sciences (LUMHS) Hospital Jamshoro was chosen. The patients’ secondary data was collected form hospital and commercial computational software MATLAB was used to carry out all the calculations and manipulations by writing a concise user defined program (code). The Autoregressive Integrated Moving Average (ARIMA) modeling approach is adopted to investigate the best forecasting model. It is found that among the various six combinations of ARIMA (p,d,q) the ARIMA (1,0,1) are the best fit models for the patients’ inflow and the patients’ admission respectively; having the lowest AIC, BIC and p-values. Since the forecast accuracy contains minimal contains minimal errors thus forecast trends show very good results. The presented procedure can be helpful to manage the patients’ volume in the hospitals and can also predict the future trend of patients’ inflow and patients’ admission with good accuracy.
creator: Sakina Kamboh
creator: Mir Ghulam Hyder Talpur
creator: Nawab Khan Chand
creator: Liaquat Ali Zardari
creator: Abdul Wasim Shaikh
creator: Shakeel Ahmed Kamboh
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-26
publication date: 2024-06-26
volume: 13
number: 2
starting page: 311
ending page: 322
doi: 10.21015/vtm.v12i1.1840

Nonparametric test for multiple crossing Survival Curves
type: item
description: Log-rank, Wilcoxon and Tarone-Ware tests are most commonly used tests for testing the overall homogeneity of survival curves, but in certain situation it appears that they have a significant loss of statistical testing power. One such case is the more than one time crossing of survival curves. The problem considered often occurs in medical research. To overcome this problem, in this article, we present and study a non-parametric test procedure based on a new weight. The proposed new weighted test has greater power to detect overall differences between more than one time crossing survival curves. Simulation studies are performed to compare the proposed method with existing methods. Furthermore, the advantage of the new test is finally exemplified in the analysis of a β-thalassaemia major data.
creator: Qamruz Zaman
creator: Nisar Ullah
creator: Syed Habib Shah
creator: Muhammad Ali
creator: Muhammad Irshad
creator: Summayyia Azam
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 349
ending page: 365
doi: 10.21015/vtm.v12i1.1839

Improved Class of Regression Type Mean Estimators in Simple Random Sampling Using Concomitant Variable: An Application in T-20 International Cricket
type: item
description: In sample surveys, the information on supplementary variables is usually used to improve the efficiency of the estimators. The ratio, product, and regression estimators are examples. This study proposes new regression estimators for population mean estimation under simple random sampling using the data of Twenty20 (T-20) International Cricket. The estimators are proposed with the help of non-conventional location parameters of concomitant variable which includes Tri-Mean, Mid-Range, and coefficient of quartile deviation. The bias and mean squared error (MSE) of new estimators have been theoretically derived up to the first-order approximation. The theoretical findings were then compared with the existing estimators numerically through simulation study and real-life data. The numerical results reveal that the suggested estimators are more efficient than the estimators in the literature in all conditions.
creator: Mujeeb Hussain
creator: Syed Habib Shah
creator: Sumayyia Azam
creator: Ikram Ullah
creator: Muhammad Irshad
creator: Nisar Ullah
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-06-30
publication date: 2024-06-30
volume: 13
number: 2
starting page: 336
ending page: 348
doi: 10.21015/vtm.v12i1.1835

Mathematical and computationalmodeling of migrations, population spread, and Location-dependent carrying capacities
type: item
description: In single-variable calculus, differential equation models for population dynamics are of high importance and by developing these ordinary differential equation (ODE) models, a practical and intelligible approach can be created to partial differential equations (PDEs). This paper researches focus on PDE models for migrations, population spread and location-dependent carrying capacities. However, logistic equation from which the PDE models develop include the diffusion equation, traveling wave equation, and building blocks with location-dependent parameters. This methodology is suitable for multivariable calculus lecture, evaluation task, and problem activity. The text is accompanied by interactive examples, and the article is formatted to be used as a CDF document in which some of the input can be hidden.
creator: Faraz Ahmed Soomro
creator: Israr Ahmed
creator: Waheed Ali Panhwar
creator: Shah Rukh Soomro
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-22
publication date: 2024-05-22
volume: 13
number: 2
starting page: 248
ending page: 263
doi: 10.21015/vtm.v12i1.1797

Efficient Class of Variance Estimators for Population using Supplementary Information in Stratified Random Sampling
type: item
description: This paper addresses an efficient class of variance estimators for population using stratified random sampling. The suggested class of estimators using supplementary information has been studied in different circumstances. The expressions of bias and mean square error (MSE) of the proposed estimators are derived up to the first degree of approximation. The theoretical comparison of the proposed and considered estimators is also discussed, which shows that the proposed estimators are more efficient than the existing estimators. Theoretical findings are validated by three different types of real data sets and simulation studies. The numerical results of the proposed and existing estimators are compared in terms of mean square error, percentage relative efficiency and diagrams. It is observed that all the proposed estimators outperform the existing estimators. For instance, the traditional unbiased estimator Ozel et.al [6] and other existing estimators. Lastly, appropriate recommendations have been provided for researchers to use these suggested estimators to solve real-world issues.
creator: Mujeeb Hussain
creator: Lakhkar Khan
creator: Qamruz Zaman
creator: Abdurrahman Sabir
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-24
publication date: 2024-05-24
volume: 13
number: 2
starting page: 264
ending page: 279
doi: 10.21015/vtm.v12i1.1794

Geometric Modelling of a Family of 4-Point Ternary Approximating Subdivision Scheme U_φ with Visual Performance
type: item
description: Making signals better than noise in communication has always been challenging for scientists. Researchers have been working on it in different ways. The computer-aided geometric design is a new research field emerging from the collaboration of computer algorithms and mathematical logic towards curve designing, in which the subdivision schemes used have a key position due to their flexible and smooth behaviour. Using parameters in these schemes allows for increased control over designing. A parameterized framework for generating a wide range of subdivision surfaces with tunable degrees of shape control is presented in the family of schemes. The properties of the proposed family make it suitable for use in isogeometric analysis, computer animation, and geometric modelling. The purpose of this paper is to construct and analyze a family of 4-point ternary subdivision schemes to smooth the curves based on the Laurant polynomial. This family is generated by tuning the weight parameter. The scheme is analysed for its different properties. The scheme has continuity. Visual performance of the subdivision scheme is also provided as an application of this proposed study.
creator: Usama Gulzar
creator: Muhammad Javed Iqbal
creator: Inayatullah Soomro
creator: Maqsood Ahmed Wassan
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-28
publication date: 2024-05-28
volume: 13
number: 2
starting page: 290
ending page: 310
doi: 10.21015/vtm.v12i1.1787

Investigation the Stochastic behaviour of the Traffic Flow: A Case Study of a Section of a Road
type: item
description: The stochastic behavior is one of the key for the current state of vehicles flow for the real time traffic behavior. This paper describe the study to investigate the stochastic behavior of real time traffic flow for a section of road using probability distribution fit over the section of road, the traffic data was collected for a week from 7:00 to 19:00 at the location Nawabshah Pakistan. The different distribution such as Normal, Lognormal, Weibull, Gamma, Exponential distribution was fit using MATLAB distribution fit by probability plot of traffic flow data. The same distribution was used for the goodness-of-fit tests by considering Kolmogorov-Smirnov, Kolmogorov-Smirnov modified, Anderson-Darling were used with p-values at 95% of confidence level and justification to accept the hypothesis test are accepted or rejects. The hypothesis accept for Normal, Weibull and Gamma distribution which accept the all hypothesis test and among these three accepted fit distribution the Normal probability distribution fit is most fitted distribution using the rank by p-value of the hypothesis tests. Keywords: Traffic flow, Goodness-of-fit, Probability Distributions, Nawabshah
creator: Mehboob Ali Jatoi
creator: Shakeel Ahmed Kamboh
creator: Oshaque Ali Abro
creator: Saeed Ahmed Rajput
creator: Liaquat Ali Zardari
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-05
publication date: 2024-05-05
volume: 13
number: 2
starting page: 189
ending page: 201
doi: 10.21015/vtm.v12i1.1784

Improved Mean Estimators for Population utilizing Dual Supplementary Characteristics under Simple Random Sampling
type: item
description: This paper makes another addition to the existing literature of population mean estimation. An improved family of mean estimators for the population is suggested using simple random sampling and utilizing the information of dual supplementary characteristics. We have conducted an extensive theoretical and numerical investigation of these recommended estimators using the criteria of bias and mean square error. The study provides a derivation of the bias and mean square error of the recommended estimators, approximating up to the first order and compared with the existing estimators. The suggested estimators are also extensively studied numerically and compared using real-world data sets and simulation studies. The theoretical and numerical comparisons show that the estimators suggested in the study are much more efficient than the competitor estimators under all situations i.e. Bahl and Tuteja [21] and other existing estimators in the literature. Therefore, the suggested family of estimators can be applied to real-life problems to obtain better results than the existing mean estimators for the population.
creator: Mujeeb Hussain
creator: Qamruz Zaman
creator: Lakhkar Khan
creator: Abdurrahman Sabir
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-07
publication date: 2024-05-07
volume: 13
number: 2
starting page: 202
ending page: 219
doi: 10.21015/vtm.v12i1.1774

A Study of Mathematical Models Used in Anaerobic Digestion of Organic Refuse
type: item
description: The mathematical computation and process evaluation of anaerobic digestion (AD) treatment of organic refuse (OR) has become quite common in process designing. Furthermore, the modelling of AD process is considered as well established and mature, and largely known as a mechanistic model structure. There are various mathematical models being used to comprehend the biochemical variations, rate of decomposition, and the methane or biogas production potentials. Moreover, these mathematical analyses of the models determine the computation of equilibrium points, helping to understand their internal stability with respect to feeding parameters, and compatibility to the output static characteristics of the entire AD process. Whereas numerical simulations are also carried out for specific biodegradation on web-based software. These simulations are useful to demonstrate the dynamic responses of the mathematical models to present the most mathematical viable solutions. However, the major issues lie in the application of AD in lab-based modelling needs improvement in characterization and the adaptation of new approaches to optimize bioenergy recoveries. Hence, this review paper discusses the selection and utilization of mathematical models for different conditions in AD treatment of OR for the best mathematical representations.
creator: Asim Ali
creator: Hareef Ahmed Keerio
creator: Oshaque Ali Abro
creator: Muhammad Noor
creator: Sallahuddin Panhwar
creator: Rasool Bux Mahar
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-20
publication date: 2024-04-20
volume: 13
number: 2
starting page: 150
ending page: 163
doi: 10.21015/vtm.v12i1.1771

Homotopy Perturbation Method with Analytics for solving Bivariate type II Fuzzy Fredholm Integral Equations
type: item
description: A numerical scheme known as homotopy perturbation method (HPM) is a powerful tool for solving a wide range of problems arising in several scientific applications. In this manuscript, we focus on bivariate type II fuzzy fredholm integral equations (BTII-FF-IEqs) to obtain fuzzy approximate solutions using HPM. The efficiency and effectiveness of the approach is tested upon numerical example and the obtained numerical results are compared with the existing exact solutions. The results reveal that the proposed method is straightforward, accurate and convenient.
creator: Sajid Hussain
creator: Shafqat Ali
creator: Abdul Salam
creator: Abid Khan
creator: Jamshaidul Hassan
creator: Hurria Ali
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-16
publication date: 2024-05-16
volume: 13
number: 2
starting page: 234
ending page: 247
doi: 10.21015/vtm.v12i1.1764

An Improved Blended Numerical Root-Solver for Nonlinear Equations
type: item
description: This study presents a novel three-step iterative approach for solving nonlinear equations in the domains of science and engineering. It represents a notable change from traditional methods like Halley by eliminating the need for second derivatives. The suggested method exhibits a sixth order of convergence and only requires five function evaluations, showcasing its efficiency with an index of roughly 1.430969. The suggested method effectively solves nonlinear problems involving equations with algebraic and transcendental terms. Comparative analysis against existing root-solving algorithms demonstrates their superior performance. The results not only confirm the strength and effectiveness of the three-step iterative approach but also highlight its potential for wide-ranging use in many scientific and technical situations.
creator: Asad Ali Chandio
creator: Asif Ali Shaikh
creator: Sania Qureshi
creator: Abdul Rehman Soomroo
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-20
publication date: 2024-04-20
volume: 13
number: 2
starting page: 164
ending page: 175
doi: 10.21015/vtm.v12i1.1763

Series Solution of Unsteady Tank Drainage for Third Order Fluid using Adomian Decomposition Method
type: item
description: In this research study the tank model of isothermal and unsteady drainage is considered. Geometry of the tank is in cylindrical and at the bottom a pipe for flow of the fluid is attached. Sub class of viscoelatic non-newtonian fluid is considered. It is planed to obtain the series solution of the model with Adomian Decompositon in the light of no slip conditon. It is also decided to compare the velocity profile of adomain decompositon with the perturbation method for this fluid. On the basis of the results and comparison efficiency of both methods will be disscussed.
creator: Shehzad Ali Malik
creator: Syed Feroz Shah
creator: Kamran Nazir Memon
creator: Afaque Ahmed Bhutto
creator: Azam Ali Amur
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-17
publication date: 2024-04-17
volume: 13
number: 2
starting page: 137
ending page: 149
doi: 10.21015/vtm.v12i1.1762

Modiﬁed Bracketing Iterative Method for Solving Nonlinear Equations
type: item
description: Non-linear equations, depicted as curves in numerous everyday situations, have long piqued the curiosity of researchers and engineers due to their frequent occurrence in practical problems. Despite attempts to tackle these equations both analytically and numerically, analytical methods often fall short when the equation's degree exceeds five, prompting the adoption of numerical approaches to yield approximate solutions. Consequently, this study places emphasis on segmenting intervals into smaller sub-intervals, with a particular focus on employing the Regula-Falsi method to integrate these segmented intervals, thereby enhancing its convergence rate. Furthermore, by utilizing the Regula-Falsi formula for interval segmentation, the number of iterations and computational time required are minimized. Additionally, the effectiveness of the proposed method is verified through numerical experiments involving various equation types, including algebraic, trigonometric, exponential, logarithmic, and transcendental equations, comparing the outcomes with established methods. The findings demonstrate that the proposed algorithm not only efficiently segments intervals but also enhances accuracy and reduces errors when these segmented intervals are utilized in conventional bracketing methods.
creator: Muhammad Imran Soomro
creator: Zubair Ahmed Kalhoro
creator: Abdul Wasim Shaikh
creator: Sanaullah Jamali
creator: Owais Ali
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-09
publication date: 2024-04-09
volume: 13
number: 2
starting page: 105
ending page: 120
doi: 10.21015/vtm.v12i1.1761

Order Structured Graphs of Cyclic Groups and their Classification
type: item
description: Let $\Gamma^{o}(G)$ with $G\cong C_{p},$ a cyclic group of order $p,$ be an order structured graph. The group $C_{p}$ will be assumed as the vertex set of the graph $\Gamma^{o}(G)$ and an edge between vertices will be built on the basis of a defined relation via order structure. Certain graphical parameters such as independence ratio, clique number, domination number, and separability are discussed. Some characterizations are proposed and proved by incorporating the defined relation. It is further proved that $\Gamma^{o}(C_{p})$ can never be a hamiltonian graph. Lastly, It is shown that $C(\Gamma^{o}(C_{p}))$ is isomorphic to $\Gamma^{o}(C_{p}).
creator: Aneela .
creator: Muhammad Khalid Mahmood
creator: Daud Ahmad
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-14
publication date: 2024-05-14
volume: 13
number: 2
starting page: 220
ending page: 233
doi: 10.21015/vtm.v12i1.1756

Patterns of Nanoporous Spherical Packing Emerging under Influence of Curvature and Confinement
type: item
description: Nanoporous membranes are popular in nanotechnology due to biomedical and industrial applications. During the past decade, experimental, theoretical and computational research into porous membranes and soft materials has opened up new mathematical dimensions. In bulk, diblock copolymers exhibit ordered morphologies such as parallel matrices of lamellae, bicontinuous matrices of gyroids, hexagonal matrices of cylinders and body-centred cubic matrices of spheres. In melt, confinement plays an essential role in tuning the frustration of the diblock copolymer system to predict more nanostructures. These nanostructures depend on the composition of the copolymers, their confining geometries and the degree of structural frustration. An isotropic 9-point stencil for Laplacian is constructed. The discrete finite-difference technique is used in polar grids to discretize the macromolecule of the diblock copolymer system to study spherical patterns to study the effect of curvature and confinement with a well-known and efficient cell dynamic simulation model. Intel FORTRAN (IFORT) codes are generated to run the CDS model and visualisation of simulation results is observed with the help of OPENDX. A comparison of the proposed study with existing experimental and computational studies is also presented.
creator: Muhammad Javed Iqbal
creator: Inayatullah Soomro
creator: Usama Gulzar
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-13
publication date: 2024-04-13
volume: 13
number: 2
starting page: 121
ending page: 136
doi: 10.21015/vtm.v12i1.1754

A versatile optional randomized response technique for use with sensitive surveys
type: item
description: In data collection from human participants, researchers in almost every survey get refusals and/or false responses from the respondents. Such refusals and false reporting are particularly common in sample surveys where the participants are asked to answer questions on sensitive topics such as cheating in examination, illegal income, marks obtained in last examination, students’ satisfaction from the teaching method, and amount of money spent on luxury items, etc. A popular approach to deal with the problem of refusals and untruthful responses is the randomized response technique. This paper introduces a randomized response model which is more precise than the available models. The proposed randomized scrambling procedure guarantees the privacy protection of the respondents for motivating them to participate in the survey.
creator: Muhammad Azeem
creator: Abdul Salam
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-04-27
publication date: 2024-04-27
volume: 13
number: 2
starting page: 176
ending page: 188
doi: 10.21015/vtm.v12i1.1750

New Results for Riemann Solution of the Cargo-LeRoux Model by the Application of Flux-Limiter Schemes
type: item
description: The Rienamm solution of the Cargo-LeRoux model has been recently introduced in [1] in which authors have found the exact solutions to the initial value problem. This work is the first attempt to apply numerical methods for the Cargo-LeRoux model. The higher-order flux limiter method applied in this paper holds the total variation diminishing property and gives smooth solutions in steep gradient regions. Various limiter functions that lead to different accuracy in numerical results are tested for the Riemann problem. The numerical investigations presented in this work can be used to review limiter-based TVD schemes extensively and to construct a class of highly efficient finite volume/ finite difference methods for such models.
creator: Sidrah Ahmed
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-05-27
publication date: 2024-05-27
volume: 13
number: 2
starting page: 280
ending page: 289
doi: 10.21015/vtm.v12i1.1748

Quadratic twist of an elliptic curve in a generalized Weierstrass equation over a function field
type: item
description: This paper mainly focuses on the construction of a quadratic twist for an elliptic curve represented in a generalized Weierstrass equation over the field Fq(t). The specific form of the quadratic twist, presented in the generalized Weierstrass equation, is determined by linear algebra approach and discussed in detail.
creator: Fida Hussain
creator: Muhamad Afzal Soomro
creator: Iqrar Ali Pali
creator: Safia Amir Dahri
creator: Abdul Rehman Soomro
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-03-25
publication date: 2024-03-25
volume: 13
number: 2
starting page: 92
ending page: 104
doi: 10.21015/vtm.v12i1.1739

Exact Solution of Tank Drainage for Bingham Fluid flow circular tube
type: item
description: Unsteady tank drainage of Bingham Plastic fluid whose flow of the fluid is due to gravity, is considered in this study. The separation of variables method is used to obtain the exact solution of the fluid. Due to very high yield stress of the fluid no slip condition is used to find out the velocity profile of the fluid. The primary objective of this work is to find the velocity profile, flow rate, shear stress, average velocity, time efflux, the relationship between the time (varying with the depth of a tank) and the time required for complete drainage from emerging partial differential equation. It has been observed that when the Bingham model parameters, flow region and pipe length increase, the time required to drain the fluid from the tank also increases. Furthermore, findings indicate that higher density and a larger pipe radius result in quicker drainage of Bingham plastic fluid from the circular tank.
creator: Ghulam Murtaza Oghai
creator: Kamran Nazir Memon
creator: Syed Feroz Shah
creator: Azam Ali Amur
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-03-23
publication date: 2024-03-23
volume: 13
number: 2
starting page: 60
ending page: 70
doi: 10.21015/vtm.v12i1.1737

Some New Generators to Produce Minimal Circular Strongly Balanced Generalized Neighbor Designs of Class-I
type: item
description: Minimal circular strongly balanced neighbor designs control the neighbor effects economically but these designs can only be constructed for v odd. For v even, their alternates are minimal circular strongly balanced generalized neighbor designs of class-I (MCSBGNDs-I) which are the designs in which (i) each treatment appears exactly once with itself as neighbors, (ii) v/2 unordered pairs appear twice as neighbors while the remaining ones appear once. On the basis of constructors developed by Munir et al. (2023), some new generators are developed in this article to generate cyclic shifts for efficient MCSBGNDs-I in blocks of (i) equal sizes, (ii) two different sizes, and (iii) three different sizes for m (mod 4) ≡ 2 & 3 with v even and k = 4l, k = 4l+2, k (odd) > 3, k (mod) ≡ 1 & k (mod) ≡ 3. Efficiency of neighbor effects and of Separability show that our proposed generators produce designs which control the neighbor effects efficiently as well as estimate the direct effects and neighbor effects independently.
creator: Abdul Salam
creator: Hurria Ali
creator: Zubia Batool Shamshi
creator: Jamshaid ul Hassan
creator: Abid Khan
creator: Sajid Hussain
rights: Copyright (c) 2024 VFAST Transactions on Mathematics
date: 2024-03-23
publication date: 2024-03-23
volume: 13
number: 2
starting page: 71
ending page: 91
doi: 10.21015/vtm.v12i1.1716