Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind

Frontiers of Mathematics in China

Volumn 7, Issue 1, 2012, Pages 69-84

  Li, Xianjuan ^a   Tang, Tao ^b  

^a FUZHOU UNIVERSITY   (China)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

Abel Volterra integral equation; convergence analysis; Jacobi spectral collocation method

Indexed keywords

EID: 84855726247     PISSN: 16733452     EISSN: 16733576     Source Type: Journal    
DOI: 10.1007/s11464-012-0170-0     Document Type: Article

References (25)

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.
3. Brunner H. Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. IMA J Numer Anal, 1986, 6: 221-239.
4. Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations Methods. Cambridge: Cambridge University Press, 2004.
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.
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.
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.
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. 3. Numer Math, 2002, 92: 289-323.
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.
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.
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.
15. Nevai P. Mean convergence of Lagrange interpolation. III. Trans Amer Math Soc, 1984, 282: 669-698.
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.
18. Ragozin D L. Constructive polynomial approximation on spheres and projective spaces. Trans Amer Math Soc, 1971, 162: 157-170.
19. p(0, b). Fract Calc & 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.
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.
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.