×

Generalized Jacobi rational spectral method on the half line. (English) Zbl 1259.65156

An orthogonal system on the half line, induced by generalized Jacobi functions is introduced. Some basic results on the generalized Jacobi rational approximation are established, which play important roles in the related spectral method. As an example of applications, the rational spectral method is proposed for partial differential equations of degenerate type. Its convergence is proved. Numerical results demonstrate its efficiency.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
41A20 Approximation by rational functions
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bernardi, C., Maday, Y.: Spectral methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol.5, Techniques of Scientific Computing, pp. 209–486. Elsevier, Amsterdam (1997)
[2] Boyd, J.P.: Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys. 69, 112–142 (1987) · Zbl 0615.65090 · doi:10.1016/0021-9991(87)90158-6
[3] Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York 1989 (2001) · Zbl 0681.65079
[4] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods, Fundamentals in Single Domains. Springer, Berlin (2006) · Zbl 1093.76002
[5] Funaro, D.: Polynomial Approximation of Differential Equations. Springer-Verlag, Berlin (1992) · Zbl 0774.41010
[6] Guo, B.-y.: Spectral Methods and Their Applications. World Scientific, Singapore (1998) · Zbl 0906.65110
[7] Guo, B.-y.: Jacobi spectral approximations to differential equations on the half line. J. Comput. Math. 18, 95–112 (2000) · Zbl 0948.65071
[8] Guo, B.-y.: Some developments in spectral methods for nonlinear partial differential equations in unbounded domains. In: Gu, C.-h., Hu, H.-s., Li, T.-t. (eds.) Differential Geometry and Related Topics, pp. 68–90. World Scientific (2002) · Zbl 1019.65080
[9] Guo, B.-y., Shen, J., Wang, L.-l.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006) · Zbl 1102.76047 · doi:10.1007/s10915-005-9055-7
[10] Guo, B.-y., Shen, J., Wang, L.-l.: Generalized Jacobi polynomials/functions and their applications. Appl. Numer. Math. 29, 1011–1028 (2009) · Zbl 1171.33006 · doi:10.1016/j.apnum.2008.04.003
[11] Guo, B.-y., Shen, J., Wang, Z.-q.: A rational approximation and its applications to differential equations on the half line. J. Sci. Comput. 15, 117–148 (2000) · Zbl 0984.65104 · doi:10.1023/A:1007698525506
[12] Guo, B.-y., Shen, J., Wang, Z.-q.: Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Int. J. Numer. Methods Eng. 53, 65–84 (2002) · Zbl 1001.65129 · doi:10.1002/nme.392
[13] Guo, B.-y., Wang, Z.-q.: Modified Chebyshev rational spectral method for the whole line. Discrete Contin. Dyn. Syst. Suppl. 365–374 (2003) · Zbl 1060.65112
[14] Guo, B.-y., Wang, Z.-q.: Legendre rational approximation on the whole line. Sci. China, Ser. A Math 47, 155–164 (2004) · Zbl 1084.41006 · doi:10.1360/04za0014
[15] Guo, B.-y., Yi, Y.-g.: Generalized Jacobi rational spectral method and its applications. J. Sci. Comput. 43, 201–238 (2010) · Zbl 1203.33013 · doi:10.1007/s10915-010-9353-6
[16] Guo, B.-y., Zhang, X.-y.: Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions. Appl. Numer. Math. 57, 455–471 (2007) · Zbl 1119.65097 · doi:10.1016/j.apnum.2006.07.032
[17] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952) · Zbl 0047.05302
[18] Maz’ja, V.G.: Sobolev Spaces. Springer-Verlag, Berlin (1985)
[19] Shen, J., Wang, L.-L.: Some recent advances in spectral methods for unbounded domains. Comm. Comp. Phys. 5, 195–241 (2009) · Zbl 1364.65265
[20] Wang, Z.-q., Guo, B.-y.: Jacobi rational approximation and spectral method for differential equations of degenerate type. Math. Comput. 77, 883–907 (2008) · Zbl 1132.41315 · doi:10.1090/S0025-5718-07-02074-1
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.