{"title":"On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities","authors":"Arvet Pedas, Mikk Vikerpuur","doi":"10.1016/j.apnum.2024.06.008","DOIUrl":null,"url":null,"abstract":"<div><p>We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>, with possible singularities of the derivatives of the solution at the left endpoint of the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo>]</mo></math></span>.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"204 ","pages":"Pages 176-187"},"PeriodicalIF":2.2000,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016892742400148X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval . The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on , with possible singularities of the derivatives of the solution at the left endpoint of the interval .
期刊介绍：
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.
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