{"title":"Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations","authors":"","doi":"10.1007/s10543-024-01011-2","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time <span> <span>\\(t=0\\)</span> </span>. A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10543-024-01011-2","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time \(t=0\). A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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