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 A hybrid collocation method for Volterra integral equations with weakly singular kernels Cao, YZ; Herdman, T; Xu, YH 2003 Source Publication SIAM JOURNAL ON NUMERICAL ANALYSIS ISSN 0036-1429 Volume 41Issue:1Pages:364-381 Abstract The 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. Keyword Volterra integral equations hybrid collocation methods weakly singular kernels Language 英语 WOS Research Area Mathematics WOS Subject Mathematics, Applied WOS ID WOS:000182450000017 Publisher SIAM PUBLICATIONS Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/18401 Collection 中国科学院数学与系统科学研究院 Corresponding Author Cao, YZ Affiliation 1.W Virginia Univ, Dept Math, Morgantown, WV 26506 USA2.Acad Sinica, Acad Math & Syst Sci, Beijing 100080, Peoples R China3.Florida A&M Univ, Dept Math, Tallahassee, FL 32307 USA4.Virginia Tech, Dept Math, Blacksburg, VA 24061 USA Recommended CitationGB/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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