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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2015

01-04-2015

On the Spectrum Computation of Non-oscillatory and Highly Oscillatory Kernel with Weak Singularity

Authors: Can Huang, Zhimin Zhang

Published in: Journal of Scientific Computing | Issue 1/2015

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Abstract

We compute the spectra of integral compact operators with weak singularity. Jacobi-spectral collocation methods are applied for problems without high oscillation. A convergence rate is obtained for general non-oscillatory operators. Furthermore, if the bilinear form associated with the kernel is positive definite, the convergence rate is doubled. A spectral Galerkin method with modified Fourier expansion is developed to compute the spectra of highly oscillatory kernel. Numerical results are presented to demonstrate the effectiveness and accuracy of our algorithms and theorems.
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Literature
1.
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)CrossRefMATH
2.
Atkinson, K.E.: The numerical solution of the eigenvalue problem for compact integral operators. Trans. Am. Math. Soc. 129, 458–467 (1967)MATH
3.
Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge University Press, Cambridge (2004)CrossRefMATH
4.
Brunner, H., Iserles, A., Norsett, S.P.: The spectral problem for a class of highly oscillatory Fredholm integral operators. IMA J. Numer. Anal. 30, 108–130 (2010)CrossRefMATHMathSciNet
5.
Brunner, H., Iserles, A., Norsett, S.P.: The computation of the spectra of highly oscillatory Fredholm integral operators. J. Integral Equ. Appl. 23, 467–518 (2011)CrossRefMATHMathSciNet
6.
Chatelin, F.: Spectral Approximation of Linear Operators. Academic Press, New York (1983)MATH
7.
Chen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)CrossRefMATHMathSciNet
8.
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Dover Publications, New York (2007)MATH
9.
Erdelyi, A.: Asymptotic Expansion. Dover Publications, Dover (1955)
10.
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals and Series. Academic Press, San Diego (2000)MATH
11.
Graham, I.: Singularity expansions for the solution of the second kind Fredholm integral equations with singular convolution kernels. J. Integral Equ. 4, 1–30 (1982)MATH
12.
Guo, B.Y., Wang, L.L.: Jacobi interpolation approximations and their applications to singular differential equations. Adv. Comput. Math. 14, 227–276 (2001)CrossRefMATHMathSciNet
13.
Hildebrand, F.B.: Introduction to Numerical Analysis. McGraw-Hill, New York (1956)MATH
14.
Huang, C., Tang, T., Zhang, Z.: Supergeometric convergence of spectral collocation methods for weakly singular Volterra and Fredholm integral equations with smooth solutions. J. Comput. Math. 29, 698–719 (2011)CrossRefMATHMathSciNet
15.
Huang, C.: Hailong Guo and Z. Zhang, A spectral collocation method for eigenvalue problems of compact integral operators. J. Integral Equ. Appl. 25, 79–101 (2013)CrossRefMATH
16.
Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal. 44, 1026–1048 (2006)CrossRefMATHMathSciNet
17.
18.
Iserles, A.: On the numerical quadrature of highly-oscillating integrals II: irregular oscillators. IMA. J. Numer. Anal. 25, 25–44 (2005)CrossRefMATHMathSciNet
19.
Iserles, A., Norsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT 44, 755–772 (2004)CrossRefMATHMathSciNet
20.
Iserles, A., Norsett, S.P.: Efficient quadrature of highly oscillatory integrals using derivatives. Proc. R. Soc. A 461, 1383–1399 (2005)CrossRefMATHMathSciNet
21.
Iserles, A., Norsett, S.P.: From high oscillation to rapid approximation I: modified Fourier expansions. IMA J. Numer. Anal. 28, 862–887 (2008)CrossRefMATHMathSciNet
22.
Iserles, A., Norsett, S.P.: From high oscillation to rapid approximation III: multivariate expansions. IMA J. Numer. Anal. 29, 882–916 (2009)CrossRefMATHMathSciNet
23.
Kaneko, H., Xu, Y.: Gaussian-type quadratures for weakly singular integrals and their applications to the fredholm integral equation of the second kind. Math. Comput. 62, 739–753 (1994)CrossRefMATHMathSciNet
24.
Levin, D.: Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comput. 38, 531–538 (1982)CrossRefMATH
25.
26.
Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications II. Springer, New York (1972)CrossRefMATH
27.
28.
29.
30.
Olver, S.: On the quadrature of multivariate highly oscillatory integrals over non-polytope domains. Numer. Math. 103, 643–665 (2006)CrossRefMATHMathSciNet
31.
Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer, New York (1996)
32.
Shen, J., Tang, T., Wang, L.L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, New York (2011)CrossRef
33.
Slater, L.J.: Confluent Hypergeometric Functions. Cambridge University Press, London (1960)MATH
34.
Vainikko, G.: Multidimensional weakly singular integral equations, Lecture Notes in Mathematics, vol. 1549. Springer, Berlin (1993)
35.
Vainikko, G., Pedas, A.: The properties of solutions of weakly singular integral equations. J. Aust. Math. Soc. (Series B) 22, 419–430 (1981)CrossRefMATHMathSciNet
36.
37.
Metadata
Title
On the Spectrum Computation of Non-oscillatory and Highly Oscillatory Kernel with Weak Singularity
Authors
Can Huang
Zhimin Zhang
Publication date
01-04-2015
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9884-3

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