{"title":"用于线性二阶边界值问题的具有自由参数的立方 B 样条修正基:在工程问题中的应用","authors":"","doi":"10.1016/j.jksus.2024.103397","DOIUrl":null,"url":null,"abstract":"<div><p>The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter <span><math><mi>γ</mi></math></span> with the purpose of modifying the weights of nearby control points. This free parameter <span><math><mi>γ</mi></math></span> can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.</p></div>","PeriodicalId":16205,"journal":{"name":"Journal of King Saud University - Science","volume":null,"pages":null},"PeriodicalIF":3.7000,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1018364724003094/pdfft?md5=00049932933731e41883f5b64e0bae13&pid=1-s2.0-S1018364724003094-main.pdf","citationCount":"0","resultStr":"{\"title\":\"A modified basis of cubic B-spline with free parameter for linear second order boundary value problems: Application to engineering problems\",\"authors\":\"\",\"doi\":\"10.1016/j.jksus.2024.103397\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter <span><math><mi>γ</mi></math></span> with the purpose of modifying the weights of nearby control points. This free parameter <span><math><mi>γ</mi></math></span> can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.</p></div>\",\"PeriodicalId\":16205,\"journal\":{\"name\":\"Journal of King Saud University - Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.7000,\"publicationDate\":\"2024-08-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S1018364724003094/pdfft?md5=00049932933731e41883f5b64e0bae13&pid=1-s2.0-S1018364724003094-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of King Saud University - Science\",\"FirstCategoryId\":\"103\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1018364724003094\",\"RegionNum\":3,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of King Saud University - Science","FirstCategoryId":"103","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1018364724003094","RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
A modified basis of cubic B-spline with free parameter for linear second order boundary value problems: Application to engineering problems
The traditional cubic B-spline method offers limited local control over the curve solution. Adjusting the position of a control point affects the entire curve, making it challenging to make localized changes, e.g., smoothness. Moreover, the basis functions vanish on one side by the cubic B-spline method near the end conditions where the initial and boundary conditions are applied. To address these limitations, this research proposes a new basis by including a free parameter with the purpose of modifying the weights of nearby control points. This free parameter can influence the curve’s behavior in specific regions as well as the entire curve. This modification of the cubic B-spline method was used to approximate the second-order derivative at each collocation point. The convergence test showed that the proposed method was second-order convergent. Numerical examples of ordinary differential equations were used with different step values to evaluate the accuracy of the proposed method. The findings persistently indicated that the proposed technique provided better error estimates as compared to the other methods discussed in the literatures.
期刊介绍:
Journal of King Saud University – Science is an official refereed publication of King Saud University and the publishing services is provided by Elsevier. It publishes peer-reviewed research articles in the fields of physics, astronomy, mathematics, statistics, chemistry, biochemistry, earth sciences, life and environmental sciences on the basis of scientific originality and interdisciplinary interest. It is devoted primarily to research papers but short communications, reviews and book reviews are also included. The editorial board and associated editors, composed of prominent scientists from around the world, are representative of the disciplines covered by the journal.