{"title":"Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel","authors":"M.H. Heydari , D. Baleanu , M. Bayram","doi":"10.1016/j.apnum.2025.06.016","DOIUrl":null,"url":null,"abstract":"<div><div>This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 355-371"},"PeriodicalIF":2.2000,"publicationDate":"2025-07-04","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/S0168927425001394","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.
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:
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