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Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. (English) Zbl 1207.65157
The authors consider Volterra integral equations of the second kind with a weakly singular kernel of the form
\[ y(t) = g(t) + \int_0^t (t-s)^{-\mu} K(t,s)y(s)\,ds,\;0<\mu<1,\quad 0\leq t\leq T, \]
and develop a Jacobi-collocation spectral method for the above integral equation. The main aim is to use Jacobi-collocation method to numerically solve Volterra integral equation on the interval \([-1,1]\). They obtain higher-order accuracy for the numerical approximation using a Jacobi spectral quadrature rule for the integral term.
Finally, numerical results are given by tables and figures. Numerical and exact solutions are compared by graphics in the \(L^{\infty}\)-norm and \(L_{w}^{\infty}\)-norm.
Reviewer: Ali Filiz (Aydin)

MSC:
65R20 Numerical methods for integral equations
45D05 Volterra integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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[1] S. BOCHKANOV AND V. BYSTRITSKY, Computation of Gauss-Jacobi quadrature rule nodes and weights, http://www.alglib.net/integral/gq/gjacobi.php
[2] Hermann Brunner, Nonpolynomial spline collocation for Volterra equations with weakly singular kernels, SIAM J. Numer. Anal. 20 (1983), no. 6, 1106 – 1119. · Zbl 0533.65087 · doi:10.1137/0720080 · doi.org
[3] Hermann Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp. 45 (1985), no. 172, 417 – 437. · Zbl 0584.65093
[4] Hermann Brunner, Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels, IMA J. Numer. Anal. 6 (1986), no. 2, 221 – 239. · Zbl 0634.65142 · doi:10.1093/imanum/6.2.221 · doi.org
[5] 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. · Zbl 1059.65122
[6] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. · Zbl 1093.76002
[7] 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.
[8] David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. · Zbl 0893.35138
[9] 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. · Zbl 0807.65141 · doi:10.1017/S0308210500028432 · doi.org
[10] 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. · Zbl 0947.47020 · doi:10.1155/S1025583499000272 · doi.org
[11] I. G. Graham and I. H. Sloan, Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in \Bbb R&sup3;, Numer. Math. 92 (2002), no. 2, 289 – 323. · Zbl 1018.65139 · doi:10.1007/s002110100343 · doi.org
[12] 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. · Zbl 1102.76047 · doi:10.1007/s10915-005-9055-7 · doi.org
[13] 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. · Zbl 0984.41004 · doi:10.1023/A:1016681018268 · doi.org
[14] 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. · Zbl 1057.41003 · doi:10.1016/j.jat.2004.03.008 · doi.org
[15] Qiya Hu, Stieltjes derivatives and \?-polynomial spline collocation for Volterra integrodifferential equations with singularities, SIAM J. Numer. Anal. 33 (1996), no. 1, 208 – 220. · Zbl 0851.65098 · doi:10.1137/0733012 · doi.org
[16] Alois Kufner and Lars-Erik Persson, Weighted inequalities of Hardy type, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. · Zbl 1065.26018
[17] Ch. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp. 45 (1985), no. 172, 463 – 469. · Zbl 0584.65090
[18] 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. · Zbl 0990.41003 · doi:10.1016/S0377-0427(00)00557-4 · doi.org
[19] Paul Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984), no. 2, 669 – 698. · Zbl 0577.41001
[20] C. K. Qu and R. Wong, Szegő’s conjecture on Lebesgue constants for Legendre series, Pacific J. Math. 135 (1988), no. 1, 157 – 188. · Zbl 0664.42012
[21] David L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Trans. Amer. Math. Soc. 150 (1970), 41 – 53. · Zbl 0208.14701
[22] David L. Ragozin, Constructive polynomial approximation on spheres and projective spaces., Trans. Amer. Math. Soc. 162 (1971), 157 – 170. · Zbl 0234.41011
[23] 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. · Zbl 0501.65062 · doi:10.1093/imanum/2.4.437 · doi.org
[24] Stefan G. Samko and Rogério P. Cardoso, Sonine integral equations of the first kind in \?_\?(0,\?), Fract. Calc. Appl. Anal. 6 (2003), no. 3, 235 – 258. · Zbl 1073.45516
[25] J. SHEN AND T. TANG, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006. · Zbl 1234.65005
[26] T. Tang, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math. 61 (1992), no. 3, 373 – 382. · Zbl 0741.65110 · doi:10.1007/BF01385515 · doi.org
[27] 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. · Zbl 0765.65126 · doi:10.1093/imanum/13.1.93 · doi.org
[28] 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. · Zbl 1174.65058
[29] T. TANG AND X. XU, Accuracy enhancement using spectral postprocessing for differential equations and integral equations, Commun. Comput. Phys., 5 (2009), pp. 779-792. · Zbl 1364.65153
[30] 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. · Zbl 1103.65089
[31] D. Willett, A linear generalization of Gronwall’s inequality, Proc. Amer. Math. Soc. 16 (1965), 774 – 778. · Zbl 0128.27604
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