Gegenbauer approximation and its applications to differential equations on the whole line.(English)Zbl 0913.41020

Author’s abstract: “A Gegenbauer approximation is discussed. Several imbedding inequalities and inverse inequalities are obtained. Some approximation results are given. By variable transformations, differential equations on the whole line are changed to certain equations on a finite interval. Gegenbauer polynomials are used for their numerical solutions. The stabilities and convergences of proposed schemes are proved. The main idea and techniques used in this paper are also applicable to other multiple-dimensional problems in unbounded domains”.

MSC:

 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 34A45 Theoretical approximation of solutions to ordinary differential equations

Keywords:

Gegenbauer approximation
Full Text:

References:

 [1] Maday, Y.; Pernaud-Thomas, B.; Vandeven, H., Une réhabilitation des méthodes spectrales de type Laguerre, Rech. Aérospat., 6, 353-379 (1985) · Zbl 0604.42026 [2] Coulaud, O.; Funaro, D.; Kavian, O., Laguerre spectral approximation of elliptic problems in exterior domains, Comp. Mech. in Appl. Mech. and Engi., 80, 451-458 (1990) · Zbl 0734.73090 [3] Funaro, D., Estimates of Laguerre spectral projectors in Sobolev spaces, (Brezinski, C.; Gori, L.; Ronveaux, A., Orthogonal Polynomials and Their Applications (1991), Scientific Publishing Co), 263-266 · Zbl 0842.46017 [4] Iranzo, V.; Falquès, A., Some spectral approximations for differential equations in unbounded domains, Comput. Methods Appl. Mech. Engrg., 98, 105-126 (1992) · Zbl 0762.76081 [5] Mavriplis, C., Laguerre polynomials for infinite-domain spectral elements, J. Comput. Phys., 80, 480-488 (1989) · Zbl 0665.65066 [6] Funaro, D.; Kavian, O., Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comp., 57, 597-619 (1990) · Zbl 0764.35007 [8] Christov, C. I., A complete orthogonal system of functions in $$L^2$$, SIAM J. Appl. Math., 42, 1337-1344 (1982) · Zbl 0562.33009 [9] Boyd, J. P., Spectral method using rational basis functions on an infinite interval, J. Comput. Phys., 69, 112-142 (1987) · Zbl 0615.65090 [10] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101 [11] Gottlieb, D.; Shu, C. W., On the Gibbs phenomenon. IV. Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp., 64, 1081-1095 (1995) · Zbl 0852.42018 [12] Bergh, J.; Löfström, J., Interpolation Spaces, an Introduction (1976), Springer-Verlag: Springer-Verlag Berlin · Zbl 0344.46071 [13] Timan, A. F., Theory of Approximation of Functions of a Real Variable (1963), Pergamon: Pergamon Oxford · Zbl 0117.29001 [14] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38, 67-86 (1982) · Zbl 0567.41008 [15] Askey, R., Orthogonal Polynomials and Spectral Functions. Orthogonal Polynomials and Spectral Functions, Regional Conference Series in Appl. Math., 21 (1975), SIAM: SIAM Philadelphia · Zbl 0298.26010 [16] Courant, R.; Friedrichs, K. O.; Levy, H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100, 32-74 (1928) · JFM 54.0486.01 [17] Richtmeyer, R. D.; Morton, K. W., Finite Difference Methods for Initial-Value Problems (1967), Interscience: Interscience New York · Zbl 0155.47502 [18] Ben-yu, Guo, A class of difference schemes of two-dimensional viscous fluid flow, TR SUST (1965) [19] Ben-yu, Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math., 163, 33-54 (1994) · Zbl 0811.65071 [20] Stetter, H. J., Stability of nonlinear discretization algorithms, (Bramble, J., Numerical Solutions of Partial Differential Equations (1966), Academic Press: Academic Press New York), 111-123 · Zbl 0149.11603
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