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A hybrid collocation method for Volterra integral equations with weakly singular kernels
Cao, YZ; Herdman, T; Xu, YH
2003
Source PublicationSIAM JOURNAL ON NUMERICAL ANALYSIS
ISSN0036-1429
Volume41Issue:1Pages:364-381
AbstractThe commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving ( nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving ( nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.
KeywordVolterra integral equations hybrid collocation methods weakly singular kernels
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000182450000017
PublisherSIAM PUBLICATIONS
Citation statistics
Cited Times:53[WOS]   [WOS Record]     [Related Records in WOS]
Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/18401
Collection中国科学院数学与系统科学研究院
Affiliation1.W Virginia Univ, Dept Math, Morgantown, WV 26506 USA
2.Acad Sinica, Acad Math & Syst Sci, Beijing 100080, Peoples R China
3.Florida A&M Univ, Dept Math, Tallahassee, FL 32307 USA
4.Virginia Tech, Dept Math, Blacksburg, VA 24061 USA
Recommended Citation
GB/T 7714
Cao, YZ,Herdman, T,Xu, YH. A hybrid collocation method for Volterra integral equations with weakly singular kernels[J]. SIAM JOURNAL ON NUMERICAL ANALYSIS,2003,41(1):364-381.
APA Cao, YZ,Herdman, T,&Xu, YH.(2003).A hybrid collocation method for Volterra integral equations with weakly singular kernels.SIAM JOURNAL ON NUMERICAL ANALYSIS,41(1),364-381.
MLA Cao, YZ,et al."A hybrid collocation method for Volterra integral equations with weakly singular kernels".SIAM JOURNAL ON NUMERICAL ANALYSIS 41.1(2003):364-381.
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