S. Jamali, Fareed Ahmed Lakho, Z. Kalhoro, Abdul Wasim Shaikh, Jinrui Guan
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.
{"title":"A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique","authors":"S. Jamali, Fareed Ahmed Lakho, Z. Kalhoro, Abdul Wasim Shaikh, Jinrui Guan","doi":"10.21015/vtm.v12i1.1712","DOIUrl":"https://doi.org/10.21015/vtm.v12i1.1712","url":null,"abstract":"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.","PeriodicalId":517760,"journal":{"name":"VFAST Transactions on Mathematics","volume":"70 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140394103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lakhkar Khan, Javid shabbir, Qazi Ubaid, Taj Farin Khan
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.
{"title":"A Refinement of Ratio Estimation in Ranked Set Sampling and Stratified Ranked Set Sampling Approaches","authors":"Lakhkar Khan, Javid shabbir, Qazi Ubaid, Taj Farin Khan","doi":"10.21015/vtm.v12i1.1619","DOIUrl":"https://doi.org/10.21015/vtm.v12i1.1619","url":null,"abstract":"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.","PeriodicalId":517760,"journal":{"name":"VFAST Transactions on Mathematics","volume":"5 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140397463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study examines a predator-prey model that includes the impact of fear and a square-root functional responseto represent herd behavior in the prey population. Our investigation aims to investigate the existence and stabilityof fixed points in this model. Through conducting an extensive analysis, we have uncovered valuable observations onthe model's behavior, namely recognizing the occurrence of period-doubling and Neimark-Sacker bifurcations.These findings provide an understanding of the intricate dynamics that govern predator-prey interactions in the presence of fear and herd behavior. We provide numerical examples to support our conclusions.
{"title":"Stability and bifurcation analysis of a discrete Leslie predator-prey model with fear effect","authors":"Naqi Abbas, R. Ahmed","doi":"10.21015/vtm.v12i1.1686","DOIUrl":"https://doi.org/10.21015/vtm.v12i1.1686","url":null,"abstract":"This study examines a predator-prey model that includes the impact of fear and a square-root functional responseto represent herd behavior in the prey population. Our investigation aims to investigate the existence and stabilityof fixed points in this model. Through conducting an extensive analysis, we have uncovered valuable observations onthe model's behavior, namely recognizing the occurrence of period-doubling and Neimark-Sacker bifurcations.These findings provide an understanding of the intricate dynamics that govern predator-prey interactions in the presence of fear and herd behavior. We provide numerical examples to support our conclusions.","PeriodicalId":517760,"journal":{"name":"VFAST Transactions on Mathematics","volume":"76 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140286141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Jamali, Z. Kalhoro, Abdul Wasim Shaikh, M. S. Chandio, Jinrui Guan
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.
{"title":"Solution of nonlinear models in engineering using a new sixteenth order scheme and their basin of attraction","authors":"S. Jamali, Z. Kalhoro, Abdul Wasim Shaikh, M. S. Chandio, Jinrui Guan","doi":"10.21015/vtm.v12i1.1624","DOIUrl":"https://doi.org/10.21015/vtm.v12i1.1624","url":null,"abstract":"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.","PeriodicalId":517760,"journal":{"name":"VFAST Transactions on Mathematics","volume":"43 48","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140400911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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