Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T11:43:55.963Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  28 November 2017

Xiong Liu  and
Yanping Chen
Xiong Liu*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China
*
*Corresponding author. Emails:yanpingchen@scnu.edu.cn (Y. P. Chen), liuxiong980211@163.com (X. Liu)
*Corresponding author. Emails:yanpingchen@scnu.edu.cn (Y. P. Chen), liuxiong980211@163.com (X. Liu)
Get access

Abstract

In this paper, a Chebyshev-collocation spectral method is developed for Volterra integral equations (VIEs) of second kind with weakly singular kernel. We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity. The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points. The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials, approximation theory for orthogonal polynomials, and the operator theory. The spectral rate of convergence for the proposed method is established in the L∞-norm and weighted L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Keywords

Chebyshev collocation methodVolterra integral equationsspectral rate of convergenceHölder continuity

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 6 , December 2017 , pp. 1506 - 1524
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brunner, H., The numerical solutions of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comput., 45 (1985), pp. 417–437.CrossRefGoogle Scholar
[2] Brunner, H., Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221–239.Google Scholar
[3] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations Methods, Cambridge University Press 2004.Google Scholar
[4] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer-Verlag 2006.CrossRefGoogle Scholar
[5] Diogo, T., McKee, S. and Tang, T., Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proceedings of The Royal Society of Edinburgh, 124A (1994), pp. 199–210.CrossRefGoogle Scholar
[6] Gogatishvill, A. and Lang, J., The generalized hardy operator with kernel and variable integral limits in Banach function spaces, J. Inequalities Appl., 4(1) (1999), pp. 1–16.Google Scholar
[7] Graham, I. G. and Sloan, I. H., Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in ℝ3 , Numer. Math., 92 (2002), pp. 289–323.Google Scholar
[8] Hu, Q., Stieltjes derivatives and polynomial spline collocation for Volterra integro-differential equa- tions with singularities, SIAM J. Numer. Anal., 33 (1996), pp. 208–220.Google Scholar
[9] Kufner, A. and Persson, L. E., Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.CrossRefGoogle Scholar
[10] Lubich, CH., Fractional linear multi-step methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45 (1985), pp. 463–469.Google Scholar
[11] Mastroianni, G. and Occorsio, D., Optimal systems of nodes for Lagrange interpolation on bounded intervals: A survey, J. Comput. Appl. Math., 134 (2001), pp. 325–341.CrossRefGoogle Scholar
[12] Nevai, P., Mean convergence of Lagrange interpolation III, Trans. Amer. Math. Soc., 282 (1984), pp. 669–698.Google Scholar
[13] Ragozin, D. L., Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc., 150 (1970), pp. 41–53.Google Scholar
[14] Ragozin, D. L., Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), pp. 157–170.Google Scholar
[15] Te Riele, H. J.J., Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA J. Numer. Anal., 2 (1982), pp. 437–449.Google Scholar
[16] Samko, S. G. and Cardoso, R. P., Sonine integral equations of the first kind in Lp(0,b), Fract. Calc. Appl. Anal., 6 (2003), pp. 235–258.Google Scholar
[17] Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[18] Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373–382.Google Scholar
[19] Tang, T., A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13 (1993), pp. 93–99.CrossRefGoogle Scholar
[20] Willett, D., A linear generalization of Gronwall's inequality, Proceedings of the American Mathematical Society, 16 (1965), pp. 774–778.Google Scholar
[21] Chen, Y., Li, X. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for weakly singular Volterra integral equation with smooth solution, J. Comput. Appl. Math., 233 (2009), pp. 938–950.Google Scholar
[22] Chen, Y., Li, X. and Tang, T., A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Math., 31(1) (2013), pp. 47–56.CrossRefGoogle Scholar
[23] Chen, Y. and Tang, T., Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with aweakly singular kernel, Math. Comput., 79 (2010), pp. 147–167.CrossRefGoogle Scholar
[24] Chen, Yanping and Gu, Zhendong, Legendre spectral-collocation method for Volterra integral differential equations with non-vanishing delay, Commun. Appl.Math. Comput. Sci., 8(1) (2013), pp. 67–98.Google Scholar
[25] Gu, Zhendong and Chen, Yanping, Chebyshev spectral-collocation method for Volterra integral equations, Contemp. Math., 586 (2013), pp. 163–170.Google Scholar
[26] Yang, Y., Chen, Y., Huang, Y. and Yang, W., Convergence analysis of Legendre-collocation methods for nonlinear volterra type integro equations, Adv. Appl. Math. Mech., 7(1) (2015), pp. 74–88.Google Scholar
[27] Yang, Yin, Chen, Yanping and Huang, Yunqing, Spectral-collocation method for fractional Fredholm integro-differential equations, J. Korean Math. Soc., 51(1) (2014), pp. 203–224.CrossRefGoogle Scholar
[28] Shi, X. and Chen, Y., Spectral-collocation method for Volterra delay integro-differential equations with weakly singular kernels, Adv. Appl. Math. Mech., 8(4) (2016), pp. 648–669.CrossRefGoogle Scholar