A hybrid collocation method for Volterra integral equations with weakly singular kernels

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	A hybrid collocation method for Volterra integral equations with weakly singular kernels
	Cao, YZ ; Herdman, T ; Xu, YH
	2003
发表期刊	SIAM JOURNAL ON NUMERICAL ANALYSIS
ISSN	0036-1429
卷号	41 期号:1 页码:364-381
摘要	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.
关键词	Volterra integral equations hybrid collocation methods weakly singular kernels
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000182450000017
出版者	SIAM PUBLICATIONS
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/18401
专题	中国科学院数学与系统科学研究院
通讯作者	Cao, YZ
作者单位	1.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
推荐引用方式 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.