Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations. (English) Zbl 0951.41006

The paper develops the spectral method for numerical solving the differential equations with singular solutions. The singularities may be not symmetric, and the main idea is to fit singular solutions by Jacobi polynomials, to compare numerical solutions with some unusual orthogonal projections of exact solutions, and to measure the errors in weighted Hilbert spaces \(L^2_\chi((-1,1))\) of functions integrated with square over \(x\in(-1,1)\) with the weight \(\chi(x)=(1-x)^\alpha(1+x)^\beta\).
The author gives several weighted inverse inequalities and considers the Jacobi approximations in the above weighted Hilbert spaces. Such approximations are just the Fourier expansion of the unknown function over Jacobi polynomials. As the applications, in the final part of the paper singular solutions for the second order linear ODE and for evolution semilinear second order PDE are constructed, and the convergence rate of the Jacobi approximations to the exact solutions is estimated.


41A10 Approximation by polynomials
46C15 Characterizations of Hilbert spaces
65L20 Stability and convergence of numerical methods for ordinary differential equations
41A25 Rate of convergence, degree of approximation
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text: DOI


[1] Kreiss, H.O.; Oliger, J., Stability of the Fourier method, SIAM J. numer. anal., 16, 421-433, (1979) · Zbl 0419.65076
[2] Gottlieb, D.; Turkel, E., On time discretization for spectral methods, Stud. appl. math., 63, 67-86, (1980) · Zbl 0453.65059
[3] Kuo, P.Y., The convergence of spectral scheme for solving two-dimensional vorticity equation, J. comput. math., 1, 353-362, (1983)
[4] Vandeven, H., Family of spectral filters for discontinuous problems, J. sci. comput., 6, 159-192, (1991) · Zbl 0752.35003
[5] Tadmor, E., Shock capturing by the spectral viscosity method, Comput. methods appl. mech. engrg., 80, 197-208, (1990) · Zbl 0729.65073
[6] Guo, B.Y., The spectral methods and their applications, (1998), World Scientific Singapore
[7] Cai, W.; Gottlieb, D.; Shu, C.W., On one-side filters for spectral Fourier approximations of discontinuous functions, SIAM J. numer. anal., 29, 905-916, (1992) · Zbl 0755.65140
[8] Cai, W.; Gottlieb, D.; Shu, C.W., Essentially nonoscillatory spectral Fourier method for shock wave calculations, Math. comp., 52, 389-410, (1989) · Zbl 0666.65067
[9] Gottlieb, D.; Shu, C.W.; Solomonoff, A.; Vandeven, O.H., On the Gibbs phenomenon. I. recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. comput. appl. math., 43, 81-98, (1992) · Zbl 0781.42022
[10] Gottlieb, D.; Shu, C.W., On the Gibbs phenomenon. II. resolution properties of Fourier methods for discontinuous waves, Comput. methods mech. engrg., 11, 27-37, (1994)
[11] Gottlieb, D.; Shu, C.W., On the Gibbs phenomenon. III. recovering exponential accuracy in a subinterval from the spectral sum of a piecewise analytic function, SIAM J. numer. anal., 33, 280-290, (1996) · Zbl 0852.42017
[12] 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 functions, Math. comp., 64, 1081-1095, (1995) · Zbl 0852.42018
[13] Gottlieb, D.; Shu, C.W., Recovering exponential accuracy from collocation point values of piecewise analytic functions, Numer. math., 71, 511-526, (1995) · Zbl 0852.42019
[14] B. Y. Guo, Gegenbauer approximation in certain Hilbert spaces and its applications to singular differential equations, SIAM J. Numer. Anal, to appear. · Zbl 0947.65112
[15] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[16] Askey, R., Orthogonal polynomials and special functions, Regional conference series in applied mathematics, 21, (1975), SIAM Philadelphia
[17] Rainville, E.D., Special functions, (1960), Macmillan New York · Zbl 0050.07401
[18] Courant, R.; Hilbert, D., Methods of mathematical physics, (1953), Academic Press New York · Zbl 0729.00007
[19] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1970), Dover New York · Zbl 0515.33001
[20] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin · Zbl 0344.46071
[21] Timan, A.F., Theory of approximation of functions of a real variable, (1963), Pergamon Oxford · Zbl 0117.29001
[22] Bernardi, C.; Maday, Y., Spectral methods, (), 209-486
[23] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge Univ. Press Cambridge
[24] Guo, B.Y., Gegenbauer approximation and its applications to differential equations on the whole line, J. math. anal. appl., 226, 180-206, (1998) · Zbl 0913.41020
[25] B. Y. Guo, Jacobi approximation and its applications to differential equations on the half line, J. Comput. Math, to appear.
[26] B. Y. Guo, Unsymmetric Jacobi approximation with applications to differential equations with rough asymptotic behaviours, unpublished manuscript.
[27] Courant, R.; Friedrichs, K.O.; Lewy, H., Über die partiellen differenzengleichungen der mathematischen physik, Math. ann., 100, 32-74, (1928) · JFM 54.0486.01
[28] Richtmeyer, R.D.; Morton, K.W., Finite difference methods for initial-value problems, (1967), Interscience New York · Zbl 0155.47502
[29] Guo, B.Y., A class of difference schemes of two-dimensional viscous fluid flow, Tr sust, (1965)
[30] Guo, B.Y., Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. math., 163, 33-54, (1994) · Zbl 0811.65071
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.