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

  • Xiong Liu (a1) (a2) and Yanping Chen (a3)

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 L 2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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Corresponding author

*Corresponding author. Emails: yanpingchen@scnu.edu.cn (Y. P. Chen), liuxiong980211@163.com (X. Liu)

References

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[1] Brunner, H., The numerical solutions of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comput., 45 (1985), pp. 417437.
[2] Brunner, H., Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221239.
[3] Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations Methods, Cambridge University Press 2004.
[4] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer-Verlag 2006.
[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. 199210.
[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. 116.
[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. 289323.
[8] Hu, Q., Stieltjes derivatives and polynomial spline collocation for Volterra integro-differential equa- tions with singularities, SIAM J. Numer. Anal., 33 (1996), pp. 208220.
[9] Kufner, A. and Persson, L. E., Weighted Inequalities of Hardy Type, World Scientific, New York, 2003.
[10] Lubich, CH., Fractional linear multi-step methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45 (1985), pp. 463469.
[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. 325341.
[12] Nevai, P., Mean convergence of Lagrange interpolation III, Trans. Amer. Math. Soc., 282 (1984), pp. 669698.
[13] Ragozin, D. L., Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc., 150 (1970), pp. 4153.
[14] Ragozin, D. L., Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc., 162 (1971), pp. 157170.
[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. 437449.
[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. 235258.
[17] Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
[18] Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373382.
[19] Tang, T., A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13 (1993), pp. 9399.
[20] Willett, D., A linear generalization of Gronwall's inequality, Proceedings of the American Mathematical Society, 16 (1965), pp. 774778.
[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. 938950.
[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. 4756.
[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. 147167.
[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. 6798.
[25] Gu, Zhendong and Chen, Yanping, Chebyshev spectral-collocation method for Volterra integral equations, Contemp. Math., 586 (2013), pp. 163170.
[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. 7488.
[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. 203224.
[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. 648669.

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