{"title":"具有时滞的奇异摄动抛物对流扩散界面问题的数值技术分析","authors":"S. Chandra Sekhara Rao, Abhay Kumar Chaturvedi","doi":"10.1080/10236198.2023.2256419","DOIUrl":null,"url":null,"abstract":"AbstractIn this article, we are interested in the numerical analysis of a singularly perturbed parabolic differential equation with time delay. The source term of the considered problem has discontinuities in the spatial variable along the interface Γd:={(d,t):t∈(0,T]}, 0<d<1, and the diffusion coefficient is a small positive perturbation parameter. The problem's solution exhibits interior and boundary layers as the perturbation parameter approaches zero, which makes the problem challenging to establish the uniform convergence of any applied classical numerical techniques with respect to the perturbation parameter. The solution to the considered problem is decomposed into regular and singular components. Some appropriate a priori bounds on derivatives of these components have been given. The domain is discretized using Shishkin mesh in the spatial direction and uniform mesh in the time direction. On the mesh points which are not on the interface, the problem is discretized using a central-difference upwind scheme. Along the interface, the problem is discretized using an especial central-difference upwind scheme that uses the average value of the source term. The ε-uniform convergence analysis of the scheme is given using the decomposition of the solution. Some numerical experiments are conducted to corroborate the efficiency of the method.Keywords: Singularly perturbeddelay differential equationsparabolic problemsconvection–diffusioninterior and boundary layersε-uniform convergenceMathematics Subject Classifications: 65M0665M12 AcknowledgmentsThe authors gratefully acknowledge the valuable comments and suggestions of the anonymous reviewers. The authors also acknowledge the IIT Delhi HPC facility for computational resources.Data availabilityData sharing is not applicable to this article as no datasets were generated or analyzed during the current study.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis research work is supported by the Science and Engineering Research Board (SERB) under Project No. MTR/2019/000614.","PeriodicalId":15616,"journal":{"name":"Journal of Difference Equations and Applications","volume":"33 1","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Analysis of a numerical technique for a singularly perturbed parabolic convection–diffusion interface problem with time delay\",\"authors\":\"S. Chandra Sekhara Rao, Abhay Kumar Chaturvedi\",\"doi\":\"10.1080/10236198.2023.2256419\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"AbstractIn this article, we are interested in the numerical analysis of a singularly perturbed parabolic differential equation with time delay. The source term of the considered problem has discontinuities in the spatial variable along the interface Γd:={(d,t):t∈(0,T]}, 0<d<1, and the diffusion coefficient is a small positive perturbation parameter. The problem's solution exhibits interior and boundary layers as the perturbation parameter approaches zero, which makes the problem challenging to establish the uniform convergence of any applied classical numerical techniques with respect to the perturbation parameter. The solution to the considered problem is decomposed into regular and singular components. Some appropriate a priori bounds on derivatives of these components have been given. The domain is discretized using Shishkin mesh in the spatial direction and uniform mesh in the time direction. On the mesh points which are not on the interface, the problem is discretized using a central-difference upwind scheme. Along the interface, the problem is discretized using an especial central-difference upwind scheme that uses the average value of the source term. The ε-uniform convergence analysis of the scheme is given using the decomposition of the solution. Some numerical experiments are conducted to corroborate the efficiency of the method.Keywords: Singularly perturbeddelay differential equationsparabolic problemsconvection–diffusioninterior and boundary layersε-uniform convergenceMathematics Subject Classifications: 65M0665M12 AcknowledgmentsThe authors gratefully acknowledge the valuable comments and suggestions of the anonymous reviewers. The authors also acknowledge the IIT Delhi HPC facility for computational resources.Data availabilityData sharing is not applicable to this article as no datasets were generated or analyzed during the current study.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis research work is supported by the Science and Engineering Research Board (SERB) under Project No. MTR/2019/000614.\",\"PeriodicalId\":15616,\"journal\":{\"name\":\"Journal of Difference Equations and Applications\",\"volume\":\"33 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2023-08-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Difference Equations and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/10236198.2023.2256419\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Difference Equations and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/10236198.2023.2256419","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Analysis of a numerical technique for a singularly perturbed parabolic convection–diffusion interface problem with time delay
AbstractIn this article, we are interested in the numerical analysis of a singularly perturbed parabolic differential equation with time delay. The source term of the considered problem has discontinuities in the spatial variable along the interface Γd:={(d,t):t∈(0,T]}, 0
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Journal of Difference Equations and Applications presents state-of-the-art papers on difference equations and discrete dynamical systems and the academic, pure and applied problems in which they arise. The Journal is composed of original research, expository and review articles, and papers that present novel concepts in application and techniques.
The scope of the Journal includes all areas in mathematics that contain significant theory or applications in difference equations or discrete dynamical systems.
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