VFAST Transactions on Mathematics
https://vfast.org/journals/index.php/VTM
<p>The <strong><em>VFAST Transactions on Mathematics</em></strong> 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.</p> <p>ISSN: 2309-0022 (Online), 2411-6343 (Print)</p> <p>Editor in Chief: Dr. Muhammad Arif, Associate Professor, Abdul Wali Khan University Mardan</p> <p><strong><span style="text-decoration: underline;"><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 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src="data:image/png;base64,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" alt="" /></p><p><span>This work is licensed under a </span><a href="http://creativecommons.org/licenses/by/3.0/" rel="license" target="_blank">Creative Commons Attribution License</a> CC BY</p><p> </p>Quadratic twist of an elliptic curve in a generalized Weierstrass equation over a function field
https://vfast.org/journals/index.php/VTM/article/view/1739
<p>This paper mainly focuses on the construction of a quadratic twist for an elliptic curve represented in a generalized Weierstrass equation over the field <span class="math math-inline"><span class="katex"><span class="katex-mathml"><em>F<sub>q</sub>(t).</em> </span></span></span> The specific form of the quadratic twist, presented in the generalized Weierstrass equation, is determined by linear algebra approach and discussed in detail.</p>Fida HussainMuhamad Afzal SoomroIqrar Ali PaliSafia Amir DahriAbdul Rehman Soomro
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-252024-03-251219210410.21015/vtm.v12i1.1739Exact Solution of Tank Drainage for Bingham Fluid flow circular tube
https://vfast.org/journals/index.php/VTM/article/view/1737
<p>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.</p>Ghulam Murtaza OghaiKamran Nazir MemonSyed Feroz ShahAzam Ali Amur
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-232024-03-23121607010.21015/vtm.v12i1.1737Some New Generators to Produce Minimal Circular Strongly Balanced Generalized Neighbor Designs of Class-I
https://vfast.org/journals/index.php/VTM/article/view/1716
<p>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.</p>Abdul SalamHurria AliZubia Batool ShamshiJamshaid ul HassanAbid KhanSajid Hussain
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-232024-03-23121719110.21015/vtm.v12i1.1716A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique
https://vfast.org/journals/index.php/VTM/article/view/1712
<p>This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.</p>Sanaullah JamaliFareed Ahmed LakhoZubair Ahmed KalhoroAbdul Wasim ShaikhJinrui Guan
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-132024-03-13121465910.21015/vtm.v12i1.1712Efficient Neighbor Designs Weakly Balanced in Circular Blocks of Three Different Sizes
https://vfast.org/journals/index.php/VTM/article/view/1687
<p>Minimal circular weakly balanced neighbor designs (MCWBNDs) are efficient to controlneighbor effects for <em>v </em>even. MCWBNDs-II are designs where3<em>v</em>/2 unordered pairs of different treatments occur twice but remaining pairs occur once as neighbors, are not available for <em>m </em>(mod 4) ≡ 0& 3 in blocks of three different sizes with <em>m </em>= (<em>v</em>-2)/2 and <em>v </em>even. Here, some new generators are presented to develop cyclic shifts to get MCWBNDs-II in blocks of three different sizes.</p>Sajid HussainJamshaid ul HassanAbid KhanHurria AliAqsa SafdarAbdul Salam
Copyright (c) 2024 VFAST Transactions on Mathematics
2023-12-312023-12-3112115517310.21015/vtm.v11i2.1687Stability and bifurcation analysis of a discrete Leslie predator-prey model with fear effect
https://vfast.org/journals/index.php/VTM/article/view/1686
<pre>This study examines a predator-prey model that includes the impact of fear and a square-root functional response<br>to represent herd behavior in the prey population. Our investigation aims to investigate the existence and stability<br>of fixed points in this model. Through conducting an extensive analysis, we have uncovered valuable observations on<br>the model's behavior, namely recognizing the occurrence of period-doubling and Neimark-Sacker bifurcations.<br>These findings provide an understanding of the intricate dynamics that govern predator-prey interactions in the <br>presence of fear and herd behavior. We provide numerical examples to support our conclusions.</pre>Naqi AbbasRizwan Ahmed
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-032024-03-03121162910.21015/vtm.v12i1.1686Morphological Investigation of Lamellae Patterns in Diblock Copolymers under Change of Thickness and Confinement in Polar Geometry
https://vfast.org/journals/index.php/VTM/article/view/1675
<p>Mathematicians collaborate computationally with practical scientists to predict the novel morphologies soft materials, having potential applications in nanotechnology. This work presents the novel lamellae morphology that the confined diblock copolymer system displayed inside the polar geometry in order to achieve this, cell dynamics simulations (CDS) model is employed on a polar mesh to study the special impacts of confinement on the self-assembly behavior of diblock copolymer systems. Under the numerical formulation of the isotropic Laplacian with polar geometry employed in the CDS model, the set of 9-point stencil is considered to discretize the Laplacian operator on polar mesh system. FTCS finite difference mechanism is taken into consideration for discretization of Laplacian operator due to its computational efficiency and accuracy. Cell dynamics simulation is efficient simulation model for large-scale simulation as compared to the other models available in literature. In the simulation, evolutionary lamella morphology is presented with and without confinement for various sizes of the circular annular pores. A comparative review of lamella morphologies achieved by simulation in comparison of experimental study is presented, coupled with a thorough investigation of symmetric and asymmetric morphologies in captivity.</p>Muhammad Javed IqbalInayatullah SoomroMaryam BibiRab Nawaz Mallah
Copyright (c) 2024 VFAST Transactions on Mathematics
2023-12-312023-12-3112117419710.21015/vtm.v11i2.1675Thermal Modeling and Performance Investigation of Proton Exchange Membrane (PEM) Fuel Cell
https://vfast.org/journals/index.php/VTM/article/view/1640
<p>Abstract This research paper presents analysis of heat generation problem in Proton Exchange Membrane (PEM) fuel cell using COMSOL Multiphysics software. PEM fuel cells are widely recognized for their high electrical power output and environmental sustainability. However, in a PEM fuel cell around 50 to 60 % of energy generated from chemical reactions is dissipated as heat energy. To address this issue PEM fuel cell stack model is designed and thermal modeling is carried out to evaluate its performance. Based on thermal modeling of surface temperature distribution of cell it is found that the cathode side of PEM fuel cell is warmer and generates more heat as compared to other parts due to the exothermic reactions,slow reaction rate,joule heating effect and material properties.<br />Moreover, it is also found that there is uniform temperature distribution across the cell due to rapid heat conduction from cathode side to the surface of the cell.The results of this study show that due to more heat generation on cathode side temperature will tend to increase.This increasing temperature enhances<br />the average cell current density but as the average cell current density increases it reduces the average cell voltage thus declining the efficiency of PEM fuel cell. Hence ,there should be an optimal temperature range between 60 to 80°C for the better performance of a PEM fuel cell. Findings of this study can serve as a valuable resource for understanding heat generation process in PEM fuel cell for the development of efficient and reliable fuel cell technology in future.</p>Ali MuradLaveet KumarKhanji HarijanHallar Parhyar
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-11-182023-11-1812111212510.21015/vtm.v11i2.1640Solution of nonlinear models in engineering using a new sixteenth order scheme and their basin of attraction
https://vfast.org/journals/index.php/VTM/article/view/1624
<p>The utilization of computers in various engineering and mathematical fields has become increasingly important in recent times. In computational and applied mathematics, computer programs can be used to efficiently process complex data in mere seconds. By utilizing different computer tools and programming languages, computers can manipulate various iterative algorithms to solve nonlinear problems. The convergence rate and computational costs per-iteration are essential factors that determine the effectiveness of an iterative algorithm. This article introduces a new and highly efficient iterative approach of sixteenth order. The authors propose a four-step iterative optimal derivative-free method for solving non-linear equations that arise in engineering application problems. The proposed method achieves an accuracy of order sixteen with only five functional evaluations, and the efficiency index is 1.741. To demonstrate the performance of the method, the authors present several numerical examples, application problems, and a study of dynamics, comparing the method with other available methods of the same order in the literature. Numerical results and dynamics are obtained using computer tools.</p>Sanaullah JamaliZubair Ahmed KalhoroAbdul Wasim ShaikhMuhammad Saleem ChandioJinrui Guan
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-012024-03-01121011510.21015/vtm.v12i1.1624Origami-Inspired Engineering: Math behind DONs
https://vfast.org/journals/index.php/VTM/article/view/1622
<p>Origami, an ancient art of paperfolding, has evolved beyond its traditional use for cultural and entertainment purposes and has found its way into artistic, scientific, and engineering realms. This research paper explores the mathematical foundations of origami, and later focuses on a specific application in biomedical engineering. This study is intended to illustrate several aspects of origami that are relevant to engineering structures, namely: geometry, pattern generation, flat-foldability, and the ability of origami tessellations to fit specific target shapes. This review further examines DNA Origami Nano-structures (DONs) which provide numerous benefits, including lower biotoxicity, increased stability, and superior adaptability, making them an excellent choice for Drug Delivery Systems.</p>Abdul Rafae Kamran
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-11-012023-11-011219311110.21015/vtm.v11i2.1622A Modified Hybrid Method For Solving Non-Linear Equations With Computational Efficiency
https://vfast.org/journals/index.php/VTM/article/view/1620
<p>This paper proposes a modified hybrid method for solving non-linear equations that improves computational efficiency while maintaining accuracy. The proposed method combines the advantages of the traditional Halley’s and mean-based methods, resulting in a more efficient algorithm. The modified hybrid method starts with Halley’s method and then switches to the mean-based method for rapid convergence. To further improve the efficiency of the algorithm, the proposed method incorporates a dynamic selection criterion to choose the appropriate method at each iteration. Numerical experiments are performed to evaluate the performance of the proposed method in comparison to other existing methods. The results show that the modified hybrid method is computationally efficient and can achieve high accuracy in a shorter time than other commonly used methods having similar features. The proposed method is applicable to a wide range of non-linear equations and can be used in various fields of science and engineering where non-linear equations arise. The modified hybrid method provides an effective tool for solving non-linear equations, offering significant improvements in computational efficiency over existing methods.</p>Shehzad Ali SoomroAsif Ali ShaikhSania QureshiBahadur Ali
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-11-252023-11-2512112613710.21015/vtm.v11i2.1620A Refinement of Ratio Estimation in Ranked Set Sampling and Stratified Ranked Set Sampling Approaches
https://vfast.org/journals/index.php/VTM/article/view/1619
<p>Estimation of population mean is a persistent subject issue in sampling surveys and many discrete efforts have been paid by various researchers to enhance the precision of the estimates by utilizing the correlated auxiliary information. In connection with this an improved version of ratio estimator are presented in this paper under the ranked set sampling scheme and stratified ranked set sampling scheme. Comparison amongst estimators is made in terms of Mean Square Errors ( ) and Percentage Relative Efficiencies ( ). The expression for of the proposed estimator is pinned-down to first order of approximations. It turns out from both simulation studies as well as real data set that the proposed estimator dominates its existing counterpart estimators.</p>lakhkar khanjavid shabbirQazi UbaidTaj Farin Khan
Copyright (c) 2024 VFAST Transactions on Mathematics
2024-03-062024-03-06121304510.21015/vtm.v12i1.1619Numerical Analysis of Inertia Effects on Pressure and Flow Patterns in Unidirectional and Reversed Newtonian Fluid Flows within a Channel
https://vfast.org/journals/index.php/VTM/article/view/1617
<p>Fluid flows in channel systems are prevalent in many industrial and engineering applications. Flow rates, Reynolds numbers, and inertia significantly impact the dynamics of fluid flows in channel systems. However, analyzing these factors' effects on fluid flows is challenging. This study investigates the impact of flow rates, Reynolds numbers, and inertia on fluid flows within a channel system filled with non-porous media using Taylor-Galerkin/Pressure-Correction Scheme. Various flow conditions, including equal and unequal flow rates, reversed flow, and unidirectional flow, are analyzed to determine the effects on pressure differences and streamline patterns by numerical simulation. The results demonstrate the crucial role of inertia on vortex development, pressure differences, and the formation of eddies in different regions of the channel. The study reveals that pressure differences increase linearly with increasing Reynolds number in unidirectional flow scenarios and that unequal flow rates can significantly alter flow behavior. The study's findings provide insight into the complex interplay between flow rates, inertia, and pressure variations in fluid flows within a channel system.</p>Rahim Bux KhokharAfaque Ahmed BhuttoIftikhar Ahmed BhuttoMuhammad Aslam SoomroUbaidullah Alias Kashif KashifIkhlaque Ahmed Bhutto
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-10-202023-10-20121426210.21015/vtm.v11i2.1617Centroidal Mean Derivative-Based Open Newton-Cotes Quadrature Rules
https://vfast.org/journals/index.php/VTM/article/view/1601
<span lang="EN-US">In this work, a new derivative-based family of open Newton-Cotes quadrature rules based on centroidal mean (CMDONC) is proposed for the estimation of definite integrals. The error terms of the modified methods are derived through the concept of precision. The local and global order of accuracy and precision for each method is also computed. The results show that the proposed methods achieve two orders of accuracy enhancement over the conventional Newton-Cotes quadrature rules (ONC). Lastly, two numerical tests are also observed which demonstrate the superiority of CMDONC rules over the classical ONC methods.</span>Sara MahesarMuhammad Mujtaba ShaikhMuhammad Saleem ChandioAbdul Wasim Shaikh
Copyright (c) 2024 VFAST Transactions on Mathematics
2023-12-312023-12-3112113815410.21015/vtm.v11i2.1601Adaptive Numerical Regularization for Variational Denoising Model with Complementary Approach
https://vfast.org/journals/index.php/VTM/article/view/1592
Denoising is a process to suppress the noise and preserve the important information in the image.<strong> </strong>In this paper, a complementary approach is proposed for variational denoising problem. A FEM (Finite Element Method) based post optimization method (mesh refinement strategy) is designed which is based on a priori estimate called mean square error. The post optimization algorithm is adaptive and intelligent in nature which allows the adaptive choice of the regularization parameters. The manual choice of the smoothing parameters is taken uniformly on spatial domain and testing of the automatic selection of these parameters in adaptive way. This is an interesting idea of computation. The intelligent and automatic choice of the values for the smoothing function is smaller in the less regular regions of the image, to refine the grid and keep constant in the other complementary regions is one of the main interests, which produces the better and enhanced version of the noisy image. The obtained results have been compared to some other methods.Mohsin Ali AmurKhuda bux Amur Azam Ali AmurIzhar Ali AmurK.N.Memon
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-11-022023-11-02121779210.21015/vtm.v11i2.1592Simulation Study on the Effect of Cover Tilt Angle of SolarStill on its Productivity
https://vfast.org/journals/index.php/VTM/article/view/1586
<p>Survival hinges on access to water, serving as both the foundation for human existence and its continuous sustenance. Developing nations grapple with the significant challenge of ensuring clean drinking water availability. One solution is the utilization of solar stills, which harness solar energy for desalination to produce potable water, all without relying on high-energy sources. Solar stills remain a viable choice for providing safe drinking water to remote regions lacking reliable energy access. In this research, a comprehensive multi-phase 3D Computational Fluid Dynamics (CFD) model was employed to investigate single-slope solar still with glass cover angles of <span dir="ltr" style="left: 173.721px; top: 447.054px; font-family: sans-serif; transform: scaleX(1.00584);" role="presentation">20</span><sup><span dir="ltr" style="left: 380.928px; top: 445.435px; font-family: monospace;" role="presentation">0</span></sup> <span dir="ltr" style="left: 389.184px; top: 447.054px; font-size: 9.70214px; font-family: sans-serif; transform: scaleX(0.97092);" role="presentation">and 25</span><sup><span dir="ltr" style="left: 421.224px; top: 445.435px; font-family: monospace;" role="presentation">0</span></sup><br />This model accurately depicts temperature variations within the solar still during different phases of operation. <br />The simulation results presented herein reveal that the efficiency is notably superior in solar stills equipped with copper plates, achieving an output of 1.24 when inclined at <span dir="ltr" style="left: 173.721px; top: 502.939px; font-family: sans-serif; transform: scaleX(0.966513);" role="presentation">20</span><sup><span dir="ltr" style="left: 330.294px; top: 501.319px; font-family: monospace;" role="presentation">0</span></sup> compared to other inclinations. It becomes evident that the tilt angle of the cover has a substantial impact on the output. Additionally, the most suitable water depth for a <span dir="ltr" style="left: 173.721px; top: 502.939px; font-family: sans-serif; transform: scaleX(0.966513);" role="presentation">20</span><sup><span dir="ltr" style="left: 330.294px; top: 501.319px; font-family: monospace;" role="presentation">0 </span></sup>angle is found to be 18mm. This cost-effective innovation is designed to provide rural populations with an efficient method to transform brackish water into potable drinking water.</p>Rajesh kumarLaveet KumarNayyar Hussain MirjatKhanji Harijan
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-10-212023-10-21121637610.21015/vtm.v11i2.1586Numerical analysis of flow rates, porous media, and Reynolds numbers affecting the combining and separating of Newtonian fluid flows
https://vfast.org/journals/index.php/VTM/article/view/1518
This study investigates the behavior of Newtonian fluids in pipes filled with and without porous media under combing and separating flow configurations. Numerical simulations are conducted to analyze the effects of changing flow rates, inertia, and porous media on flow patterns, vortex development, and pressure difference. The aim of this study is to examine the impact of inertia on flow behavior by analyzing the streamline patterns, vortex growth, and intensity at different Reynolds numbers, ranging from Re=1 to Re=5000.Rahim Bux KhokharAfaque Ahmed BhuttoNoor Fatima SiddiquiFozia ShaikhIftikhar Ahmed Bhutto
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2023-06-302023-06-3012121723610.21015/vtm.v11i1.1518Dynamic complexity of a discrete predator-prey model with prey refuge and herd behavior
https://vfast.org/journals/index.php/VTM/article/view/1512
This paper examines a discrete predator-prey model's complex dynamics. Using bifurcation and center manifold theory, we study period-doubling and Neimark-Sacker bifurcations at positive fixed points and their direction. Numerical simulations confirm the theoretical conclusions that the model's dynamics rely on Euler method step size. The model's behavior is also affected by the prey population's conservation rate. The model suggests that excessive conservation may reduce predator populations, causing food shortages. Thus, predator-prey dynamics management must account for prey conservation rateMuhammad Asim ShahzadRizwan Ahmed
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2023-06-202023-06-2012119421610.21015/vtm.v11i1.1512Effects of Heat Transfer on Flow of MHD Maxwell Nanofluid on Stretching and Shrinking Surfaces.
https://vfast.org/journals/index.php/VTM/article/view/1498
<p>In present article, there is studied heat, and the mass transfer characteristics of boundary layer flow of the Maxwell-nanofluid across stretching/shrinking surface along chemical reactions, transverse magnetic field and thermal radiation. Applying similarity transformation, the system of governing nonlinear PDEs are reduced into form of ODEs. The achieved equations are solved with help of bvp4c in Matlab computer software. The impacts of specified parameters include, suction parameter, Deborah number, magnetic parameter, thermophoresis parameter, chemical reaction parameter, Schmidt number, Brownian motion parameter, Prandlt number and thermal slip parameter are examined on velocity, temperature and nanoparticles concentration fields (profiles). Moreover, skin friction, Nusselt number and Sherwood number are achieved at numerous values of applied parameters which are demonstrated through graphs. Some of the key findings show that an increase in suction increases skin friction, Nusselt number and Shewood number along the variation of the stretching/shrinking parameter An increase in thermophoresis, magnetic and Brownian motion parameters increase the temperature fields of Maxwell nanofluid, while Deborah number, Prandtl number, suction parameter and thermal slip parameter decrease it.</p><p><strong><br /></strong></p>Ibrahim DayoSayed Feroz ShahFozia ShaikhSanjay Kumar
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2023-06-092023-06-0912118019410.21015/vtm.v11i1.1498Average Error Based Adaptive Regularization Control For The Gradient Constancy Variational Stereo Model
https://vfast.org/journals/index.php/VTM/article/view/1492
<p>The study of the stereo vision problems is most crucial and challenging task in image processing and computer vision. The Stereo Vision problems address the investigation of the correspondence between the two images of same scene (stereo pairs) captured from two different views. Generally, these problems are inverse and ill posed. To deal with such problems the energy-based regularization techniques are considered as an efficient and most successful approaches. The adaptive finite element method is used here as discretization method for the partial differential equation obtained from the optimization of the designed energy functional. Such type of the regularization generally depends on the smoothness parameters, and their suitable choice. The choice of the smoothing parameter in an adaptive way and specifically their choice as a scalar function over the whole computational domain is an interesting idea. In this work, a variational model based on the gradient constancy assumption is proposed, moreover a post optimization method (mesh refinement strategy) is designed which is based on a priori estimate called average absolute disparity error estimate. The post optimization is based on an adaptive intelligent algorithm which is efficient in identifying the less regular regions of the computed disparity image and reduces the value of the parameter to refine the grid. Consequently, the smoothness appears in the solution which is the main goal.</p>Izhar Ali AmurKhuda Bux AmurMuzaffar Bashir ArainMohsin Ali AmurK.N Memon
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2023-06-062023-06-0612115616910.21015/vtm.v11i1.1492Numerical Simulation of 2D Shallow water equation with constant external body force by using Finite difference method
https://vfast.org/journals/index.php/VTM/article/view/1469
In this paper the numerical modeling and simulation of 2D shallow water equations is discussed with the non-flat topography. The sets of these equations is solved by means of the Crank-Nicolson finite difference method with constant external body force and Darcy Weisbach equation is used for friction slope parameter. We have obtained the important results that, as soon as we start the time then Height evaluation function has the maximum amplitude wave length which is decreasing when the time increases. The numerical solution algorithm works well and enables to predict the water elevation and velocity at any instance and any location in the domain.Saeed Ahmed RajputShakeel Ahmed KambohKhuda Bux AmurSaifullah MemonAbbas Ali Ghoto
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2023-06-072023-06-0712117017910.21015/vtm.v11i1.1469An Algorithm Coded with R to Generate GN2 -designs in Circular Blocks
https://vfast.org/journals/index.php/VTM/article/view/1465
<p>Minimal neighbor balanced designs are economical, therefore, these are preferred by the experimenters to minimize the bias due to neighbor effects. Minimal circular balanced neighbor designs cannot be constructed for almost every case of <em>v </em>even, where <em>v </em>is number of the treatments to be compared. For <em>v </em>even, the circular GN<sub>2</sub>-designs in which each treatment appears exactly once as neighbors with all other treatments except the one with which it appears twice, are considered the better alternate to the minimal balanced neighbor designs. In this article, an algorithm is developed to generate the circular GN<sub>2</sub>-designs for <em>v</em> even which can be converted directly into minimal circular balanced and strongly balanced neighbor designs. This algorithm is also coded with R.</p>Javid ShabbirJamshaid UL HassanMahmood UL HassanKhadija NoreenSajid HussainRashid Ahmed
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2023-09-072023-09-07121162710.21015/vtm.v11i2.1465MHD flow and heat transfer of micropolar nanofluid on a linearly stretching/shrinking porous surface
https://vfast.org/journals/index.php/VTM/article/view/1456
<p>In this paper, there is considered incompressible steady two-dimensional laminar MHD boundary layer flow, heat and mass transfer characteristics of micropolar nanofluid across a linearly stretching/shrinking porous surface. The effects of the magnetic, thermal slip, mass slip and heat source sink parameters are also considered. By applyingn appropriate similarity variables, the system of governing partial differential equations associated to micropolar nanofluid flow is transformed into a system of non - linear ordinary differential equations. The resulting equations are numerically solved in the Maple software by using shooting technique. The impact of the different applied parameters on skin friction, couple stress, Nusselt and the Sherwood numbers along the related profiles are determined for both stretching and shrinking cases of the surfaces. It was observed that with an increase in suction and magnetic parameters, the fluid velocity decreased. An increment in the thermal slip, the fluid temperature decreased and nanoparticles concentration decreases as the mass slip parameter is enhanced. An increase in concentration decreases but temperature increases. While, concentration and temperature both increase due to increase in thermophoresis parameter, and concentration also increases by increase in rate of chemical reaction. Thus, suction at the boundary and magnetic parameter acted as flow controlling parameter. It is believed that this type of investigation is very much helpful for the manufacturing of complex fluids and also for cleaning oil from surfaces.</p>Sanjay KumarAsif Ali ShaikhHazoor Bux LanjwaniSayed Feroz Shah
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2023-05-092023-05-0912114115410.21015/vtm.v11i1.1456The effect of uniform and exponential streams on Magnetohydrodynamic flows of viscous fluids.
https://vfast.org/journals/index.php/VTM/article/view/1442
<p>In this research, we explore the equations of motion governing the behaviour of a finitely conducting incompressible fluid with variable viscosity, heat transfer, and the presence of a transverse magnetic field. By solving these equations, we obtain exact solutions that describe flows in which the vorticity distribution is varying indirect proportion to the stream function perturbed by both uniform and exponent streams. To obtain these exact solutions, we introduce a transformation variable that allows us ttransforming a non-linear system into a linear form the governing equations. From there, we are able to derive several exact solutions that provide insight into the behaviour of the fluid in these conditions.</p>Iftikhar Ahmed BhuttoAfaque Ahmed BhuttoRahim Bux KhokharMuhammad Aslam SoomroFozia Shaikh
Copyright (c) 2023 VFAST Transactions on Mathematics
2023-04-292023-04-2912112114010.21015/vtm.v11i1.1442Cubic Iterated Methods of Numerical Differential Method for Solving Non-Linear Physical Functions
https://vfast.org/journals/index.php/VTM/article/view/1420
In this research two iterated methods have been developed for solving non-linear equations, which arises in applied sciences and engineering. The proposed iterated methods are converged cubically, and it is derived from modified euler method and improved euler method with steffensen method. The cubic iterated methods of numerical differential method are work on physical application functions and compared with variant newton iterated method. The numerical outcome of proposed cubic iterated methods of numerical differential method is examined with C++/MATLAB. From the numerical results, it can be observed that the cubic iterated methods of numerical differential method are good for accuracy point of view, iteration perception and function evaluation as the assessment of variant newton iterated method for solving non-linear physical functions.Umair Khalid QureshiMuzaffar B. AarainRahim Bux KhokharManzar Bashir
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2023-04-132023-04-131219610610.21015/vtm.v11i1.1420Development of an Explicit Iterative Numerical Scheme Over the Modified Euler’s Method
https://vfast.org/journals/index.php/VTM/article/view/1419
<p>The main theme of this research paper is to propose an explicit iterative numerical scheme over the modified Euler’s method (MEM) for solving initial value problems in ordinary differential equations. This iterative scheme has been proposed by changing the inner slope of the MEM by the mean of main slope of MEM and slope of explicit forward Euler’s method. The proposed iterative scheme has been tested for its performance by analyzing different numerical factors related to numerical algorithm and found it consistent, more accurate with the order of convergence 2 and it also shows greater stability region as compared to existing methods which are compared with proposed method.</p>Tuljaram MeghwarPrem KumarRahim Bux KhokharAsif Ali ShaikhEvren HincalMuhammad Anwar Solangi
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2023-04-192023-04-1912110712010.21015/vtm.v11i1.1419Simulation of 13 points Laplacian operator in cylindrical mesh system by using explicit finite difference technique
https://vfast.org/journals/index.php/VTM/article/view/1414
In nearest period of time a considerable and remarkable work have been done on the graphic design artificial intelligence, optimization, solving numerical approximation of ordinary as well as Partial differential equation by some differential operator, Discrete Laplacian operator is one of the most modern technique used for discretizatise the Ordinary differential equations (ODES) and Partial Differential equation (PDES). It is most simple technique, the researcher have taken great interest, as they may be able to find the novelty in their results specially in simulation of di-block co-polymer, solving the partial differential equation and also in finding light equation, wave equation in one as well as two dimension and etc. Discrete laplacian operator plays an essential role in the field of science and technology, shape interpolation, smoothing of the surface area, editing the meshes are the best usages of discrete laplacian operator In this study, a finite difference scheme is applied to develop the Laplacian operator using a cylindrical mesh system containing thirteen-point stencils. Laplacian operator has a significant role in the study of numerous mathematical and scientific phenomena. This employed in numerous significant computational and mathematical models, such as the Cell Dynamic Simulation system for the study of cutting-edge materials and image processing. The laplacian operator is necessary for being an isotropic discrete differential operator in a variety of models. When computing discrete isotropic results on curved surfaces, the curvilinear coordinate system may be a good option.Rabnawaz MallahInayatullah SoomroAltaf AhmedDost MuhammadSarang LatifIrshad Ali
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2023-03-312023-03-31121849510.21015/vtm.v11i1.1414To Investigate Obstacle Configuration Effect on Vortex Driven Combustion Instability
https://vfast.org/journals/index.php/VTM/article/view/1411
Combustion instability inside a chamber may lead to catastrophic failure. It is due to inappropriate combustion or flow physics. Flow-driven instability is mostly governed by surface, corner, or obstacle vortex shedding. Inhibitors<br />are placed inside a solid fuel combustion chamber to control the burning of the solid fuel grain. They burn slowly as compared to solid fuel and create protrusions inside the combustion chamber. These protrusions work as flow obstacles. An obstacle vortex is shed from the inhibitor and produces pressure oscillations. The effects of inhibitor position on pressure oscillation inside a solid fuel combustion chamber is investigated in the present study. Large eddies must be resolved to compute vortex-driven flow. Therefore, the Detached Eddy Simulation is applied to cylindrical combustion chambers having inhibitor and nozzle. Five different configurations are simulated. All parameters of different configurations are similar except inhibitor position. The inhibitor is moved upstream and downstream from the reference position to examine its effect on pressure oscillations. Pressure time histories at eight distinct places of the combustion chamber are recorded. Fast Fourier Transform (FFT) has been utilized to get pressure oscillation frequency and amplitude. It is interesting to reveal that the maximum amplitude of pressure oscillation occurs when the inhibitor to stagnation point distance<br />is close to the combustion chamber diameter. It is up to 70% higher as compare to reference inhibitor position results. Computed results are also compared with available experimental data for validation purpose.Afaque Ahmed BhuttoSyed Feroz ShahRahim Bux KhokharKhanji HarijanMukkarum Hussain
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2023-03-292023-03-29121678210.21015/vtm.v11i1.1411The effect of oscillating streams on heat transfer in viscous magnetohydrodynamic MHD fluid flow
https://vfast.org/journals/index.php/VTM/article/view/1386
This study focuses on developing and proving exact solutions for equations of motion involving a fluid with finite conductivity, variable viscosities, and heat transfer in the presence of a transverse magnetic field. By utilizing a transformation variable, the governing equation is transformed into a assortment of simple ordinary differential equations, enabling accurate solutions to be achieved for the problem. The solutions demonstrate that the distribution of vorticity is proportional to the stream function, which is disturbed by oscillating (sine or cosine) or even and exponential streams. This study compares the profiles of steady fluid flow with changing viscosity and heat transfer travelling on a plane. The comparison is used to identify differences in the profiles of providing insight into the changes that occur in interesting factors. These findings have significant implications for understanding fluid dynamics in complex systems and may have important applications in fields such as engineering and physics.Afaque Ahmed BhuttoIftikhar AhmedSaeed Ahmed RajputSyed Asad Raza Shah
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2023-03-052023-03-0512111610.21015/vtm.v11i1.1386Exact solution on the impact of slip condition for unsteady tank drainage flow of Ellis fluid
https://vfast.org/journals/index.php/VTM/article/view/1352
<p>In this paper, we look into the effect of slip condition on isothermal and incompressible Ellis fluid of an unsteady tank drainage flow. The non-linear PDE (partial differential equation) is solved exactly by applying the governing continuity and momentum equations, subject to the proper boundary condition, using the separation of variables approach. Unique situations this model put out by Ellis fluid is used to develop concepts like Newtonian, Power law model, and Bingham Plastic model solution. On setting the slip parameterexact solution for Ellis fluid flow is retrivred as well as Newtonian solution is bring back, which was done through Bernoulli's equation. Expressions for velocity field, pipe shear stress, volume flux, velocity average, depth of fluid in the tank at different times and also the relationship between length of the time be different with depth of the tank and the length of time required to complete the drainage is determined. Graphical representation is given of the effects of various development factors on the velocity field <em>Vz</em> and fluid depth <em>H(t</em>). The tank can empty faster for Ellis fluid compared to its special situations, according to the analogy of Ellis, Power law, Newtonian, and Binghan plastic fluids for the relation of depth with respect to time.<strong></strong></p>Naina Salar ShaikhKamran Nazir MemonMuhammad Suleman SialA. M. Siddiqui
Copyright (c) 2023 VFAST Transactions on Mathematics
2022-12-312022-12-3112120121210.21015/vtm.v10i2.1352Improved Mid-Point Derivative based Closed Newton-Cotes Quadrature Rule
https://vfast.org/journals/index.php/VTM/article/view/1344
The main motivation for this work lies in the construction of new and efficient methods to improve the efficiency index of ‘Mid-Point Derivative Based Closed Newton-Cotes Quadrature Rules’. Proposed methods use the derivative values at the mid-points in each strip of integrations such as Mid-Point Derivative Based Closed Trapezoidal, Simpson One Third, Simpson Three eight and Bool’s Rule. The degree of precision and order of accuracy of proposed methods are higher than all the existing methods. Furthermore, error terms of the proposed methods are calculated by using the concept of precision. An extensive comparison of the proposed formulas, classical and mid-point quadrature rules for the number of function evaluation, error terms, coefficient of error terms and results obtained from some different problems are given. The comparisons illustrate that the new proposed Closed Newton-Cotes Rules are much superior to Classical Rules and Zhao and Li’s ‘Mid-Point Derivative Based Closed Newton-Cotes Quadrature’ schemes.Mir Sarfraz Khalil SaandShakeel RindZuabir AhmedAbdul WasimOwais Ali
Copyright (c) 2023 VFAST Transactions on Mathematics
2022-12-312022-12-3112121322910.21015/vtm.v10i2.1344Effect of Slip Condition on Unsteady Tank Drainage Flow of third Order Fluid
https://vfast.org/journals/index.php/VTM/article/view/1338
Unsteady, isothermal, and incompressible third-order fluid is considered in the tank, cylindrical coordinates are taken into account and the flow of the fluid is considered in the z-direction. The behavior of viscoelastic fluid is analyzed with slip conditions. Velocity profile, flow rate, and mathematical relation between time depth of mass and time efflux, etc. are calculated with the perturbation method. Perturbation is a suitable technique for nonlinear partial differential equations because it converts them into the linear form which is easy to solve. Different physical flow parameters are inspected graphically. A comparison of slip and no-slip is presented also Newtonian and third-order fluids are compared too.Muneer Ahmed MaharKamran Nazir MemonSyed Feroz ShahAzam Ali AmurA. M. Siddiqui
Copyright (c) 2023 VFAST Transactions on Mathematics
2022-12-312022-12-3112118920010.21015/vtm.v10i2.1338The melting behavior of Paraffin RT-50 in a finned cylindrical surface
https://vfast.org/journals/index.php/VTM/article/view/1320
<p>The energy provision is one of the main concerns of modern technological processes and thermal management systems. Through latent heat energy, the storage of thermal energy using phase-change materials is examined in this paper. Paraffin Rubitherm 50 is filled in the cylinder. The base of the cylinder is heated and the vertical surface is made adiabatic. The melting procedure for two cases namely the plane surface and finned surface of the cylinder are considered. The melt fractions are observed and photographed for fixed intervals of time from solid state to total melt state. Initially, the melting of specified PCM was slow and then it became faster when convection heat transfer is accompanied with the conduction. The melting of PCM is geared with fin presence.</p>Waris AliAsif Ali ShaikhFeroz ShahSajjad Hussain
Copyright (c) 2023 VFAST Transactions on Mathematics
2022-12-312022-12-3112117518810.21015/vtm.v10i2.1320Analysis of Two-Level Complex Shifted Laplace Preconditioner and Deflation-Based Preconditioner for Helmholtz Equation
https://vfast.org/journals/index.php/VTM/article/view/1304
A Long time deflation preconditioner is used to speed up the convergence of the Krylov subspace method. The discretization of Helmholtz equation with Dirichlet boundary condition by finite difference method obtained any linear system. Resolving a large wavenumber requires a larger number of Grid points, i.e. large linear systems. Thus due to the large linear system, many (sparse) direct methods have taken more memory, So we have used the (iterative technique) Krylov subspace method. One of the problems of the Krylov subspace method is the required preconditioner for better convergence. We use (CSLP) as a preconditioner and drive eigenvalues of (CSLP). However, with increasing wavenumber CSLP shows slow convergence behavior. Then we use another projection-type preconditioner as a deflation preconditioner. A rigorous Fourier analysis (RFA) is a separate research idea to examine the con- vergence of the iterative method included in this article. We analyze the deflation preconditioner with a complex shifted Laplace preconditioner (CSLP) which exhibition spectral behavior of the preconditioner, which is favorable to the Krylov method.Rao Faisal RajputHanan ShiekhK. B. Amur
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2022-12-312022-12-3112114716310.21015/vtm.v10i2.1304Solution of Time-Fractional Third-Order Partial Differential Equations of One and Higher Dimensions
https://vfast.org/journals/index.php/VTM/article/view/1301
The purpose of this study is to develop the third order time fractional partial differential equations (PDEs) in one and higher dimensions, by taking Laplace Adomian decomposition method (LADM) and q-homotopy analysis transform method (q-HATM). To define fractional derivative, the Caputo operator is used for both fractional and integer orders. The solutions are obtained in the form of series. To understand the procedure of the suggested procedure, three numerical examples are taken. The graphs are plotted for the proposed solution at different values of fractional order ???? which is 0< ???? ≤ 1. Both proposed methods are implemented by using (LADM) and (q-HATM) showing that the proposed technique is found to be better and accurate instrument for solving linear and non-linear time fractional PDEs. The Novelty of the proposed study is that the provided solution for fractional order partial differential equations has never been attempted for third order, this means that the provided solution can solve the third order and could be generalized for the higher order also.Safdar AliFozia HanifMuhammad IlyasRehan ShamsMuhammad RehanSyed Inayatullah
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2023-03-172023-03-17121285110.21015/vtm.v10i2.1301A new three step derivative free method using weight function for numerical solution of non-linear equations arises in application problems
https://vfast.org/journals/index.php/VTM/article/view/1289
Abstract In this paper a three-step numerical method, using weight function, has been derived for finding the root of non-linear equations. The proposed method possesses the accuracy of order eight with four functional evaluations.<br />The efficiency index of the derived scheme is 1.682. Numerical examples, application problems are used to demonstrate the performance of the presented schemes and compare them to other available methods in the literature of the same order. Matlab, Mathematica 2021 & Maple 2021 software were used for numerical results.Sanaullah JamaliZubair Ahmed KalhoroAbdul Wasim ShaikhMuhammad Saleem ChandioSanaullah Dehraj
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2022-12-312022-12-3112116417410.21015/vtm.v10i2.1289A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function
https://vfast.org/journals/index.php/VTM/article/view/1288
It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in literature to find root but their cost are very high In this regard we introduce a two-step fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods.Sanaullah JamaliZubair Ahmed KalhoroAbdul Wasim ShaikhMuhammad Saleem ChandioSanaullah Dehraj
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2022-12-312022-12-3112113714610.21015/vtm.v10i2.1288An Innovative Soft Rough Dual Hesitant Fuzzy Sets and Dual Hesitant Fuzzy Soft Rough Sets
https://vfast.org/journals/index.php/VTM/article/view/1219
This article seeks to demonstrate the novel properties of soft rough dual hesitant fuzzy sets (DHFSRSs) and dual hesitant fuzzy soft rough sets (SRDHFSs). The fundamental characteristics of DHFSRSs and SRDHFSs are thoroughly<br />investigated. Additionally, we present a portrayal hypothesis for the DHFSRSs and SRDHFSs, which demonstrates that the level arrangements of the DHFSRSs and SRDHFSs can be used to characterize both the lower and upper DHFSRSs and SRDHFSs estimates in an identical manner.Tasawar AbbasRehan ZafarSana AnjumAmbreen AyubZamir Hussain
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2023-03-162023-03-16121172710.21015/vtm.v11i1.1219Unsteady Flow of an MHD Tangent Hyperbolic Nanofluid Over a Stretching Sheet
https://vfast.org/journals/index.php/VTM/article/view/1215
Abstract The recent article addresses the unsteady flow of MHD incompressible tangent hyperbolic fluid with Nanofluid particles in the direction of a stretching surface. Nano-fluid is related to thermo-phoretic and Brownian movement. With proper help through the transformation procedure, the set of non-linear (PDEs) is re-framed into (ODEs). The initiate expressions of momentum, temperature field, and nano-particle concentration are composed into groups of nonlinear equations. That consequential terminology is computed shooting system. The impact of fundamental parameters on the flow field, thermal circulation, and meditation is described. Moreover, the flow field behavior due to the Wall friction, local Nusselt, and Sherwood numbers are examined. This study is significant as this transformation determined the shooting technique’s numerical result and ensured the physical parameters’ behavior graphically. The results show that the velocity field diminishes by escalating the Weissenberg (We) figure and power-law index (n), while thermal and concentration fields remain to detect elevating at similar parameters. Furthermore, the computed result is compared with existing literature and gets accuracy.Muhammad Asif JamalM. FaizanAhmed FaridFozia ShaikhFozia hanif
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2022-06-302022-06-30121405610.21015/vtm.v10i1.1215Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation
https://vfast.org/journals/index.php/VTM/article/view/1211
In this study the effect of the coecients on the convergence of numerical solution of general second order linear homogeneous partial differential equation has been investigated. The main objective was to determine the sensitivity of the coecients of the PDE in relation to the domain and mesh size. The nite difference method was used to discretize the PDE and numerical solution was obtained by implementing the algorithm on MATLAB. The outcomes of the research have provided interesting facts about the stable values of the coecients. From the results it is found that the arbitrary coecients d and e are more sensitive as compared to a, b, c and f. The outcomes of this research study are expected to provide the ways to predict and control the numerical solution convergence behavior obtained by the general second order PDE based on the variable coecients of the PDE.Liaquat Ali ZardariShakeel Ahmed KambohAbbas Ali GhotoDr. Kirshan Kumar LuhanaDr. Shah Zaman Nizamani
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2022-12-092022-12-0912110211710.21015/vtm.v10i2.1211