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Sobolev orthogonal Legendre rational spectral methods for exterior problems. (English) Zbl 1499.65701

Summary: The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
41A20 Approximation by rational functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
35J25 Boundary value problems for second-order elliptic equations
35K10 Second-order parabolic equations
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[1] Boyd, J. P., Spectral methods using rational basis functions on an infinite interval, J. Comput. Phys., 69, 112-142 (1987) · Zbl 0615.65090
[2] Boyd, J. P., Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70, 63-88 (1987) · Zbl 0614.42013
[3] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover, New York · Zbl 0994.65128
[4] Christov, C. I., A complete orthogonal system of functions in \(####\) space, SIAM J. Appl. Math., 42, 1337-1344 (1982) · Zbl 0562.33009
[5] Coulaud, O.; Funaro, D.; Kavian, O., Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Methods Appl. Mech. Eng., 80, 451-458 (1990) · Zbl 0734.73090
[6] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods, Fundamentals in Single Domains (2006), Springer: Springer, Berlin · Zbl 1093.76002
[7] Guo, B. Y.; Shen, J., On spectral approximations using modified Legendre rational functions: application to Korteweg-de Vries equation on the half line, Indiana J. Math., 50, 181-204 (2001) · Zbl 0992.65111
[8] Guo, B. Y.; Shen, J.; Xu, C. L., Generalized Laguerre approximation and its applications to exterior problems, J. Comp. Math., 23, 113-130 (2005) · Zbl 1073.65130
[9] Guo, B. Y.; Wang, L. L.; Wang, Z. Q., Generalized Laguerre interpolation and pseudospectral method for unbounded domains, SIAM J. Numer. Anal., 43, 2567-2589 (2006) · Zbl 1116.41002
[10] Liu, F. J.; Li, H. Y.; Wang, Z. Q., Spectral methods using generalized Laguerre functions for second and fourth order problems, Numer. Algor., 75, 1005-1040 (2017) · Zbl 1385.65048
[11] Liu, F. J.; Wang, Z. Q.; Li, H. Y., A fully diagonalized spectral method using generalized Laguerre functions on the half line, Adv. Comput. Math., 43, 1227-1259 (2017) · Zbl 1387.76077
[12] Li, S.; Li, Q. L.; Wang, Z. Q., Sobolev orthogonal Legendre rational spectral methods for problems on the half line, Math. Methods Appl. Sci., 43, 255-268 (2020) · Zbl 1451.65211
[13] Meddahi, S.; Márquez, A., A combination of spectral and finite elements for an exterior problem in the plane, Appl. Numer. Math., 43, 275-295 (2002) · Zbl 1015.65061
[14] Singh, H., Approximate solution of fractional vibration equation using Jacobi polynomials, Appl. Math. Comput., 317, 85-100 (2018) · Zbl 1426.74161
[15] Singh, H.; Srivastava, H. M., Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients, Phys. A, 523, 1130-1149 (2019) · Zbl 07563443
[16] Shen, J.; Wang, L. L., Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57, 710-720 (2007) · Zbl 1118.65111
[17] Shen, J.; Tang, T.; Wang, L. L., Spectral Methods: Algorithms, Analysis and Applications (2011), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1227.65117
[18] Singh, H.; Ghassabzadeh, F. A.; Tohidi, E.; Cattani, C., Legendre spectral method for the fractional Bratu problem, Math. Methods Appl. Sci., 43, 5941-5952 (2020) · Zbl 1451.65162
[19] Wang, Z. Q.; Guo, B. Y., Jacobi rational approximation and spectral method for differential equations of degenerate type, Math. Comp., 77, 883-907 (2008) · Zbl 1132.41315
[20] Wu, J. X.; Wang, Z. Q., Mixed Fourier-generalized Jacobi rational spectral method for two-dimensional exterior problems, Inter. J. Numer. Anal. Modeling, 10, 657-672 (2013) · Zbl 1281.65132
[21] Wang, Z. Q.; Guo, B. Y.; Zhang, W., Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions, J. Comput. Appl. Math., 217, 277-298 (2008) · Zbl 1144.65077
[22] Zhang, R.; Wang, Z. Q.; Guo, B. Y., Mixed Fourier-Laguerre spectral and pseudospectral methods for exterior problems using generalized Laguerre functions, J. Sci. Comput., 36, 263-283 (2008) · Zbl 1203.65271
[23] Wang, Z. Q.; Guo, B. Y.; Wu, Y. N., Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains, Disc. Cont. Dyn. Sys. B, 11, 1019-1038 (2009) · Zbl 1169.65113
[24] Zhang, X. Y.; Guo, B. Y., Spherical harmonic-generalized Laguerre spectral method for exterior problems, J. Sci. Comput., 27, 523-537 (2006) · Zbl 1099.76048
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