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Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

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Front Math Chin    2012, Vol. 7 Issue (1) : 69-84    https://doi.org/10.1007/s11464-012-0170-0
RESEARCH ARTICLE
Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind
Xianjuan LI1, Tao TANG2()
1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China; 2. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
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Abstract

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ?(t, s) = (t - s)-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938-950], the error analysis for this approach is carried out for 0<μ<1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

Keywords Jacobi spectral collocation method      Abel-Volterra integral equation      convergence analysis     
Corresponding Author(s): TANG Tao,Email:ttang@math.hkbu.edu.hk   
Issue Date: 01 February 2012
 Cite this article:   
Xianjuan LI,Tao TANG. Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind[J]. Front Math Chin, 2012, 7(1): 69-84.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0170-0
https://academic.hep.com.cn/fmc/EN/Y2012/V7/I1/69
1 Ali I, Brunner H, Tang T. A spectral method for pantograph-type delay differential equations and its convergence analysis. J Comput Math , 2009, 27: 254-265
2 Ali I, Brunner H, Tang T. Spectral methods for pantograph-type differential and integral equations with multiple delays. Front Math China , 2009, 4(1): 49-61
doi: 10.1007/s11464-009-0010-z
3 Brunner H. Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J Numer Anal , 1986, 6: 221-239
doi: 10.1093/imanum/6.2.221
4 Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge: Cambridge University Press, 2004
doi: 10.1017/CBO9780511543234
5 Canuto C, Hussaini M Y, Quarteroni A, Zang T A. Spectral Methods Fundamentals in Single Domains. Berlin: Springer-Verlag, 2006
6 Chen Y, Tang T. Spectral methods for weakly singular Volterra integral equations with smooth solutions. J Comput Appl Math , 2009, 233: 938-950
doi: 10.1016/j.cam.2009.08.057
7 Chen Y, Tang T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math Comp , 2010, 79: 147-167
doi: 10.1090/S0025-5718-09-02269-8
8 Diogo T, McKee S, Tang T. Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proc Roy Soc Edinburgh Sext A , 1994, 124: 199-210
doi: 10.1017/S0308210500028432
9 Gogatishvill A, Lang J. The generalized hardy operator with kernel and variable integral limits in Banach function spaces. J Inequal Appl , 1999, 4(1): 1-16
10 Graham I G, Sloan I H. Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ?3. Numer Math , 2002, 92: 289-323
doi: 10.1007/s002110100343
11 Hesthaven J S. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J Numer Anal , 1998, 35: 655-676
doi: 10.1137/S003614299630587X
12 Kufner A, Persson L E. Weighted Inequalities of Hardy Type. New York: World Scientific, 2003
13 Lubich Ch. Fractional linear multi-step methods for Abel-Volterra integral equations of the second kind. Math Comp , 1985, 45: 463-469
doi: 10.1090/S0025-5718-1985-0804935-7
14 Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey. J Comput Appl Math , 2001, 134: 325-341
doi: 10.1016/S0377-0427(00)00557-4
15 Nevai P. Mean convergence of Lagrange interpolation. III. Trans Amer Math Soc , 1984, 282: 669-698
doi: 10.1090/S0002-9947-1984-0732113-4
16 Quarteroni A, Valli A. Numerical Approximation of Partial Differential Equations. Berlin: Springer-Verlag, 1997
17 Ragozin D L. Polynomial approximation on compact manifolds and homogeneous spaces. Trans Amer Math Soc , 1970, 150: 41-53
doi: 10.1090/S0002-9947-1970-0410210-0
18 Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc , 1971, 162: 157-170
19 Samko S G, Cardoso R P. Sonine integral equations of the first kind in Lp(0, b). Fract Calc &amp; Appl Anal , 2003, 6: 235-258
20 Shen J, Tang T. Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006
21 Shen J, Tang T, Wang L-L. Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics , Vol 41. Berlin: Springer, 2011
22 Tang T. Superconvergence of numerical solutions to weakly singular Volterra integrodifferential equations. Numer Math , 1992, 61: 373-382
doi: 10.1007/BF01385515
23 Tang T. A note on collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J Numer Anal , 1993, 13: 93-99
doi: 10.1093/imanum/13.1.93
24 Tang T, Xu X, Cheng J. On spectral methods for Volterra type integral equations and the convergence analysis. J Comput Math , 2008, 26: 825-837
25 Wan Z, Chen Y, Huang Y. Legendre spectral Galerkin methods for second-kind Volterra integral equations. Front Math China , 2009, 4(1): 181-193
doi: 10.1007/s11464-009-0002-z
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