×

zbMATH — the first resource for mathematics

Generalized Jacobi polynomials/functions and their applications. (English) Zbl 1171.33006
For arbitrary real numbers \(\alpha,\,\beta\) the authors define the generalized Jacobi polynomials/functions \(j_n^{\alpha,\beta}\) by \[ j_n^{\alpha,\beta}(x)=\omega^{\hat{\alpha},\hat{\beta}}(x)\,J_{n_1}^{\tilde{\alpha},\tilde{\beta}}(x), \quad n\geq n_0^{\alpha,\beta},\;-1<x<1, \] where \[ \hat{\alpha}=\begin{cases} -\alpha,&\alpha\leq -1\\ 0,&\alpha>-1\end{cases}; \quad \tilde{\alpha}=\begin{cases}-\alpha,&\alpha\leq -1\\ \alpha &\alpha>-1\end{cases},\quad \omega^{a,b}(x)=(1-x)^a(1+x)^b \] and \[ n_0=n_0^{\alpha,\beta}:=[\hat{\alpha}]+[\hat{\beta}],\;n_1=n_1^{\alpha,\beta}:=n-n_0^{\alpha,\beta}. \] (\([\cdot]\) the greatest integer function)
For \(\alpha,\,\beta>-1\) and for \(\alpha,\,\beta\) negative integers they are polynomials of degree \(n\), coinciding—up to a multiplicative constant—with the definition in G. Szegő [Orthogonal Polynomials, AMS (1975; Zbl 0305.42011)].
Other values of the parameters lead to \[ j_n^{\alpha,\beta}(x)=\begin{cases}(1-x)^{-\alpha}\,(1+x)^{-\beta}\,J_{n_1}^{-\alpha,-\beta}(x), & \alpha,\beta\leq -1;\;n_1=n-[-\alpha]-[-\beta] \\ (1-x)^{-\alpha}\,J_{n_1}^{-\alpha,\beta}(x),&\alpha\leq -1,\beta > -1;\;n_1=n-[-\alpha]\\ (1+x)^{-\beta}\,J_{n_1}^{\alpha,-\beta}(x),&\alpha>-1,\beta\leq -1;\;n_1=n-[-\beta].\end{cases} \] The GJP/Fs satisfy orthogonality properties, a Sturm-Liouville equation, derivative recurrence relations, approximation properties on the underlying Sobolev space, etc.
As an application spectral Galerkin methods for higher order differential equations are studied, including error estimates and, furthermore, some numerical results are given for the equation \[ u^{(6)}(x)-u(x)=f(x) \] on \((-1,1)\) with boundary conditions \(u(\pm 1),u'(\pm 1),u''(\pm 1)\) and driving force \(f(x)\) such that the exact solution is \[ u(x)=(1-x)e^x \] and \[ u(x)=(1+x)^pe^x. \]

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R., Boundary value problems for higher ordinary differential equations, (1986), World Scientific Singapore
[2] Askey, R., Orthogonal polynomials and special functions, (1975), Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0298.26010
[3] Babuška, I.; Guo, B., Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions, Numer. math., 85, 2, 219-255, (2000) · Zbl 0970.65117
[4] Babuška, I.; Guo, B., Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spaces. I. approximability of functions in the weighted Besov spaces, SIAM J. numer. anal., 39, 5, 1512-1538, (2001/2002), (electronic) · Zbl 1008.65078
[5] Bergh, J.; Löfström, J., Interpolation spaces, an introduction, (1976), Springer-Verlag Berlin · Zbl 0344.46071
[6] Bernardi, C.; Dauge, M.; Maday, Y., Spectral methods for axisymmetric domains, (1999), Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier Paris
[7] Bernardi, C.; Maday, Y., Approximations spectrales de problèmes aux limites elliptiques, (1992), Springer-Verlag Paris · Zbl 0773.47032
[8] C. Bernardi, Y. Maday, Basic results on spectral methods. R94022, Univ. Pierre et Marie Curie, Paris, 1994
[9] Bernardi, C.; Maday, Y., Spectral method, part 2, () · Zbl 1004.65119
[10] Bona, J.L.; Dougalis, V.A.; Karakashian, O.A., Conservative, high-order numerical schemes for the generalized KdV-type equation, Philos. trans. roy. soc. lond. ser. A, 351, 107-164, (1995) · Zbl 0824.65095
[11] Boutayeb, A.; Twizell, E., Numerical methods for the solution of special sixth-order boundary value problems, Int. J. comput. math., 50, 207-233, (1992) · Zbl 0773.65055
[12] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Publications Inc. Mineola, NY · Zbl 0987.65122
[13] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods: fundamentals in single domains, (2006), Springer-Verlag Heidelberg · Zbl 1093.76002
[14] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. comp., 38, 67-86, (1982) · Zbl 0567.41008
[15] Courant, R.; Hilbert, D., Methods of mathematical physics. volume 1, (1953), Interscience Publishers New York · Zbl 0729.00007
[16] Dey, B.; Khare, A.; Kumar, C.N., Stationary solitons of the fifth order KdV-type equations and their stability, Phys. lett. A, 223, 449-452, (1996) · Zbl 1037.35502
[17] El-Gamel, M.; Cannon, J.R.; Zayed, A.I., Sinc – galerkin method for solving linear sixth-order boundary value problems, Math. comp., 73, 1-19, (2004)
[18] Fornberg, B., A pseudospectral fictitious point method for high order initial-boundary value problems, SIAM J. sci. comput., 28, 5, 1716-1729, (2006), (electronic) · Zbl 1123.65104
[19] Funaro, D., Polynomial approximations of differential equations, (1992), Springer-Verlag Berlin · Zbl 0785.65087
[20] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, (1977), SIAM-CBMS Philadelphia · Zbl 0412.65058
[21] Guo, B., Gegenbauer approximation and its applications to differential equations on the whole line, J. math. anal. appl., 226, 180-206, (1998) · Zbl 0913.41020
[22] Guo, B., Spectral methods and their applications, (1998), World Scientific Publishing Co. Inc. River Edge, NJ
[23] Guo, B., Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. math. anal. appl., 243, 373-408, (2000) · Zbl 0951.41006
[24] Guo, B., Jacobi spectral approximation and its applications to differential equations on the half line, J. comput. math., 18, 95-112, (2000) · Zbl 0948.65071
[25] Guo, B.; Shen, J.; Wang, L.-L., Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. sci. comp., 27, 305-322, (2006) · Zbl 1102.76047
[26] Guo, B.; Wang, L., Jacobi approximations and jacobi – gauss-type interpolations in nonuniformly weighted Sobolev spaces, J. approx. theory, 128, 1-41, (2004)
[27] Huang, W.Z.; Sloan, D.M., The pseudospectral method for third-order differential equations, SIAM J. numer. anal., 29, 6, 1626-1647, (1992) · Zbl 0764.65058
[28] Kichenassamy, S.; Olver, P.J., Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. math. anal., 23, 5, 1141-1166, (1992) · Zbl 0755.76023
[29] Merryfield, W.J.; Shizgal, B., Properties of collocation third-derivative operators, J. comput. phys., 105, 1, 182-185, (1993) · Zbl 0767.65074
[30] Parkes, E.J.; Zhu, Z.; Duffy, B.R.; Huang, H.C., Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV-type equations, Phys. lett. A, 248, 219-224, (1998)
[31] Rainville, E.D., Special functions, (1960), Macmillan New York · Zbl 0050.07401
[32] Shen, J., Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials, SIAM J. sci. comput., 15, 1489-1505, (1994) · Zbl 0811.65097
[33] Shen, J., A new dual-petrov – galerkin method for third and higher odd-order differential equations: application to the KDV equation, SIAM J. numer. anal., 41, 1595-1619, (2003) · Zbl 1053.65085
[34] Shen, J.; Wang, L.-L., Legendre and Chebyshev dual-petrov – galerkin methods for hyperbolic equations, Comput. methods appl. mech. engrg., 196, 37-40, 3785-3797, (2007) · Zbl 1173.65342
[35] Szegö, G., Orthogonal polynomials, Amer. math. soc. collog. publ., vol. 23, (1975), Amer. Math. Soc. Providence, RI · JFM 65.0278.03
[36] Timan, A.F., Theory of approximation of functions of a real variable, (1963), Pergamon Oxford · Zbl 0117.29001
[37] Zabusky, N.J.; Galvin, C.J., Shallow water waves, the korteveg – de-Vries equation and solitons, J. fluid mech., 47, 811-824, (1971)
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.