Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel

Chen, Yanping; Tang, Tao

doi:10.1090/S0025-5718-09-02269-8

Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel
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by Yanping Chen and Tao Tang;

Math. Comp. 79 (2010), 147-167

DOI: https://doi.org/10.1090/S0025-5718-09-02269-8

Published electronically: June 16, 2009

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Abstract:

In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval $[-1,1]$, so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the $L^{\infty }$-norm and the weighted $L^2$-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

References

S. Bochkanov and V. Bystritsky, Computation of Gauss-Jacobi quadrature rule nodes and weights, http://www.alglib.net/integral/gq/gjacobi.php
Hermann Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal. 20 (1983), no. 6, 1106–1119. MR 723827, DOI 10.1137/0720080
Hermann Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp. 45 (1985), no. 172, 417–437. MR 804933, DOI 10.1090/S0025-5718-1985-0804933-3
Hermann Brunner, Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels, IMA J. Numer. Anal. 6 (1986), no. 2, 221–239. MR 967664, DOI 10.1093/imanum/6.2.221
Hermann Brunner, Collocation methods for Volterra integral and related functional differential equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004. MR 2128285, DOI 10.1017/CBO9780511543234
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. MR 2223552
Y. Chen and T. Tang, Convergence analysis for the Chebyshev collocation methods to Volterra integral equations with a weakly singular kernel, submitted to SIAM J. Numer. Anal.
David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980, DOI 10.1007/978-3-662-03537-5
Teresa Diogo, Sean McKee, and T. Tang, Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 199–210. MR 1273745, DOI 10.1017/S0308210500028432
A. Gogatishvili and J. Lang, The generalized Hardy operator with kernel and variable integral limits in Banach function spaces, J. Inequal. Appl. 4 (1999), no. 1, 1–16. MR 1733113, DOI 10.1155/S1025583499000272
I. G. Graham and I. H. Sloan, Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in $\Bbb R^3$, Numer. Math. 92 (2002), no. 2, 289–323. MR 1922922, DOI 10.1007/s002110100343
Ben-Yu Guo, Jie Shen, and Li-Lian Wang, Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comput. 27 (2006), no. 1-3, 305–322. MR 2285783, DOI 10.1007/s10915-005-9055-7
Guo Ben-yu and Wang Li-lian, Jacobi interpolation approximations and their applications to singular differential equations, Adv. Comput. Math. 14 (2001), no. 3, 227–276. MR 1845244, DOI 10.1023/A:1016681018268
Ben-yu Guo and Li-lian Wang, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory 128 (2004), no. 1, 1–41. MR 2063010, DOI 10.1016/j.jat.2004.03.008
Qiya Hu, Stieltjes derivatives and $\beta$-polynomial spline collocation for Volterra integrodifferential equations with singularities, SIAM J. Numer. Anal. 33 (1996), no. 1, 208–220. MR 1377251, DOI 10.1137/0733012
Alois Kufner and Lars-Erik Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1982932, DOI 10.1142/5129
Ch. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp. 45 (1985), no. 172, 463–469. MR 804935, DOI 10.1090/S0025-5718-1985-0804935-7
G. Mastroianni and D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey, J. Comput. Appl. Math. 134 (2001), no. 1-2, 325–341. MR 1852573, DOI 10.1016/S0377-0427(00)00557-4
Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
C. K. Qu and R. Wong, Szegő’s conjecture on Lebesgue constants for Legendre series, Pacific J. Math. 135 (1988), no. 1, 157–188. MR 965689
David L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 41–53. MR 410210, DOI 10.1090/S0002-9947-1970-0410210-0
David L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Trans. Amer. Math. Soc. 162 (1971), 157–170. MR 288468, DOI 10.1090/S0002-9947-1971-0288468-1
Herman J. J. te Riele, Collocation methods for weakly singular second-kind Volterra integral equations with nonsmooth solution, IMA J. Numer. Anal. 2 (1982), no. 4, 437–449. MR 692290, DOI 10.1093/imanum/2.4.437
Stefan G. Samko and Rogério P. Cardoso, Sonine integral equations of the first kind in $L_p(0,b)$, Fract. Calc. Appl. Anal. 6 (2003), no. 3, 235–258. MR 2035650
J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math. 61 (1992), no. 3, 373–382. MR 1151776, DOI 10.1007/BF01385515
T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal. 13 (1993), no. 1, 93–99. MR 1199031, DOI 10.1093/imanum/13.1.93
Tao Tang, Xiang Xu, and Jin Cheng, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math. 26 (2008), no. 6, 825–837. MR 2464738
T. Tang and X. Xu, Accuracy enhancement using spectral postprocessing for differential equations and integral equations, Commun. Comput. Phys., 5 (2009), pp. 779-792.
Zheng-su Wan, Ben-yu Guo, and Zhong-qing Wang, Jacobi pseudospectral method for fourth order problems, J. Comput. Math. 24 (2006), no. 4, 481–500. MR 2243117
D. Willett, A linear generalization of Gronwall’s inequality, Proc. Amer. Math. Soc. 16 (1965), 774–778. MR 181726, DOI 10.1090/S0002-9939-1965-0181726-3