{"title":"A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions","authors":"Sekar Elango , L. Govindarao , R. Vadivel","doi":"10.1016/j.apnum.2024.09.001","DOIUrl":null,"url":null,"abstract":"<div><p>This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"207 ","pages":"Pages 323-338"},"PeriodicalIF":2.2000,"publicationDate":"2024-09-13","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/S0168927424002320","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
This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.
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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