×

A fast solver for the Hilbert-type singular integral equations based on the direct Fourier spectral method. (English) Zbl 1285.65085

Summary: We develop a fast and direct Fourier spectral method for solving the Hilbert-type singular integral equation. This method leads to a fully discrete linear system, whose coefficient matrix is expressed as the sum of a sparse matrix and a quasi-circulant matrix. We show that it requires a nearly linear computational cost to obtain and then solve the fully discrete linear system. We also prove that the proposed algorithm preserves the optimal convergent order. One numerical experiment is presented to demonstrate its approximate accuracy and computational efficiency, verifying the theoretical estimates.

MSC:

65R20 Numerical methods for integral equations
65F50 Computational methods for sparse matrices
65T50 Numerical methods for discrete and fast Fourier transforms
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen, Z.; Zhou, Y., An efficient algorithm for solving Hilbert type singular integral equations of the second kind, Comput. Math. Appl., 58, 632-640 (2009) · Zbl 1189.65308
[2] Du, J., On the numerical solution for singular integral equations with Hilbert kernel, J. Comput. Math., 2, 148-166 (1989) · Zbl 0687.65125
[3] Du, J., On the collocation methods for singular integral equations with Hilbert kernel, Math. Comp., 78, 891-928 (2009) · Zbl 1198.65054
[4] Golberg, M. A., Numerical Solution of Integral Equations (1990), Plenum Press: Plenum Press New York · Zbl 0735.65092
[5] Knapek, S., Hyperbolic cross approximation of integral operators with smooth kernel. Technical Report 665, SFB 256 Univ. Bonn (2000)
[6] Krenk, S., Numerical quadrature of periodic singular integral equations, J. Inst. Math., 21, 181-187 (1978) · Zbl 0376.65045
[7] Loakimidis, N. I., A natural interpolation formula for the numerical solution of singular integral equation with Hilbert kernel, BIT, 23, 92-104 (1983) · Zbl 0505.65067
[8] Muskhelishvili, N. I., Singular Integral Equations (1968), Noordhoff: Noordhoff Groningen · Zbl 0174.16202
[9] Saranen, J.; Vainikko, G., Periodic Integral and Pseudodifferential Equations with Numerical Approximation (2002), Springer: Springer Berlin · Zbl 0991.65125
[10] Saranen, J.; Vainikko, G., Fast solution of integral and pseudodifferential equations on closed curves, Math. Comp., 67, 1473-1491 (1998) · Zbl 0907.65138
[11] Saranen, J.; Vainikko, G., Two-grid solutions of Symm’s integral equation, Math. Nachr., 177, 265-279 (1996) · Zbl 0858.65135
[12] Saranen, J.; Vainikko, G., Trigonometric collocation methods with product integration for boundary integral equations on closed curves, SIAM J. Numer. Anal., 33, 1577-1596 (1996) · Zbl 0855.65119
[13] Zhao, X., The Galerkin method for solving singular integral equation with Hilbert kernel, Acta Math. Sci., 16, 249-257 (1996) · Zbl 0945.65524
[14] Cai, H.; Xu, Y., A fast Fourier-Galerkin method for solving singular boundary integral equations, SIAM J. Numer. Anal., 48, 1965-1984 (2008) · Zbl 1175.65140
[15] Cai, H., An efficient algorithm for solving the generalized airfoil equation, J. Sci. Comput., 37, 99-114 (2008)
[16] Cai, H., A fast Petrov-Galerkin method for solving the generalized airfoil equation, J. Complexity, 25, 420-436 (2009) · Zbl 1172.76038
[17] Cai, H., A fast solver for a hypersingular boundary integral equation, Appl. Numer. Math., 59, 1960-1969 (2009) · Zbl 1171.65085
[18] Cai, H., A fast numercial solution for the first kind boundary integral equation for the Helmholtz equation, BIT (2012)
[19] Jiang, Y.; Xu, Y., Fast Fourier-Galerkin methods for solving singular boundary integral equations: numerical integration and precondition, J. Comput. Appl. Math., 234, 2792-2807 (2010) · Zbl 1193.65206
[20] Wang, B.; Wang, R.; Xu, Y., Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs, Sci. China Ser. A: Math., 53, 1-22 (2010) · Zbl 1193.65232
[21] Kress, R., Linear Integral Equations (1989), Springer-Verlag: Springer-Verlag New York
[22] Atkinson, K. E., The Numerical Solution of Integral Equations of Second Kind (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0899.65077
[23] Baszenski, G.; Delvos, F. J., A discrete Fourier transform scheme for Boolean sums of trigonometric operators, (Chui, C. K.; Schempp, W.; Zeller, K., Multivariate Approximation Theory IV. Multivariate Approximation Theory IV, ISNM 90 (1989), Birkhauser: Birkhauser Basel), 15-24 · Zbl 0689.42011
[24] Iserles, A., On the numerical quadrature of highly oscillating integrals I: Fourier transforms, IMA J. Numer. Anal., 24, 365-391 (2004) · Zbl 1061.65149
[25] Iserles, A., On the numerical quadrature of highly oscillating integrals II: Irregular oscillators, IMA J. Numer. Anal., 25, 25-44 (2005) · Zbl 1069.65148
[26] Iserles, A.; Nøsett, S. P., On quadrature methods for highly oscillatory integrals and their implementation, BIT, 44, 755-772 (2004) · Zbl 1076.65025
[27] Jiang, Y.; Xu, Y., Fast discrete algorithms for sparse Fourier expansions of high dimensional functions, J. Complexity, 26, 21-51 (2010) · Zbl 1184.65124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.