{"title":"涉及奇点的时间分数次扩散方程有效数值格式的比较研究","authors":"Bappa Ghosh, Jugal Mohapatra","doi":"10.1007/s40009-024-01488-z","DOIUrl":null,"url":null,"abstract":"<div><p>This article compares two efficient numerical schemes for solving time-fractional subdiffusion equations. The fractional derivative is taken in the Caputo sense of order <span>\\(\\alpha \\in (0,1)\\)</span>. The solution to this problem generally has a layer due to mild singularity near <span>\\(t=0\\)</span>. As a result, the standard numerical scheme degrades the convergence rate on uniform meshes. The L1-2 and L2-<span>\\(1_{\\sigma }\\)</span> techniques are used in the time direction on a graded mesh to study the layer behavior of the solution. In contrast, the spatial derivative is approximated by applying the central finite difference formula on a uniform mesh. The computational results and comparison with existing literature demonstrate the effectiveness of the proposed schemes.</p></div>","PeriodicalId":717,"journal":{"name":"National Academy Science Letters","volume":"48 4","pages":"487 - 491"},"PeriodicalIF":1.3000,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Comparative Study of Efficient Numerical Schemes for Time-Fractional Subdiffusion Equation Involving Singularity\",\"authors\":\"Bappa Ghosh, Jugal Mohapatra\",\"doi\":\"10.1007/s40009-024-01488-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This article compares two efficient numerical schemes for solving time-fractional subdiffusion equations. The fractional derivative is taken in the Caputo sense of order <span>\\\\(\\\\alpha \\\\in (0,1)\\\\)</span>. The solution to this problem generally has a layer due to mild singularity near <span>\\\\(t=0\\\\)</span>. As a result, the standard numerical scheme degrades the convergence rate on uniform meshes. The L1-2 and L2-<span>\\\\(1_{\\\\sigma }\\\\)</span> techniques are used in the time direction on a graded mesh to study the layer behavior of the solution. In contrast, the spatial derivative is approximated by applying the central finite difference formula on a uniform mesh. The computational results and comparison with existing literature demonstrate the effectiveness of the proposed schemes.</p></div>\",\"PeriodicalId\":717,\"journal\":{\"name\":\"National Academy Science Letters\",\"volume\":\"48 4\",\"pages\":\"487 - 491\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"National Academy Science Letters\",\"FirstCategoryId\":\"4\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40009-024-01488-z\",\"RegionNum\":4,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"National Academy Science Letters","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1007/s40009-024-01488-z","RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
A Comparative Study of Efficient Numerical Schemes for Time-Fractional Subdiffusion Equation Involving Singularity
This article compares two efficient numerical schemes for solving time-fractional subdiffusion equations. The fractional derivative is taken in the Caputo sense of order \(\alpha \in (0,1)\). The solution to this problem generally has a layer due to mild singularity near \(t=0\). As a result, the standard numerical scheme degrades the convergence rate on uniform meshes. The L1-2 and L2-\(1_{\sigma }\) techniques are used in the time direction on a graded mesh to study the layer behavior of the solution. In contrast, the spatial derivative is approximated by applying the central finite difference formula on a uniform mesh. The computational results and comparison with existing literature demonstrate the effectiveness of the proposed schemes.
期刊介绍:
The National Academy Science Letters is published by the National Academy of Sciences, India, since 1978. The publication of this unique journal was started with a view to give quick and wide publicity to the innovations in all fields of science