zbMATH — the first resource for mathematics

A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations. (English) Zbl 1219.65077
Summary: This paper analyzes a method for solving the third- and fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov-Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials \(P_n^{\alpha,\beta}\) with \(\alpha ,\beta \in ( - 1,\infty \)) and \(n\) is the polynomial degree. By choosing appropriate base functions, the resulting system is sparse and the method can be implemented efficiently. A Jacobi-Jacobi dual-Petrov-Galerkin method for the differential equations with variable coefficients is developed. This method is based on the Petrov-Galerkin variational form of one Jacobi polynomial class, but the variable coefficients and the right-hand terms are treated by using the Gauss-Lobatto quadrature form of another Jacobi class. Numerical results illustrate the theory and constitute a convincing argument for the feasibility of the proposed numerical methods.

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI
[1] Szegö, G., Orthogonal polynomials, Amer. math. soc. colloq. publ., 23, (1985) · JFM 65.0278.03
[2] Gottlieb, D.; Shu, C.-W., On the Gibbs phenomenon and its resolution, SIAM rev., 29, 644-668, (1997) · Zbl 0885.42003
[3] D. Tchiotsop, D. Wolf, V. Louis-Dorr, R. Husson, Ecg data compression using Jacobi polynomials, in: Proceedings of the 29th Annual International Conference of the IEEE EMBS, 2007, pp. 1863-1867.
[4] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. algorithms, 42, 137-164, (2006) · Zbl 1103.65119
[5] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of the integrated forms for second-order equations using ultraspherical polynomials, Anziam j., 48, 361-386, (2007) · Zbl 1138.65104
[6] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. numer. math., 58, 1224-1244, (2008) · Zbl 1152.65112
[7] Doha, E.H.; Bhrawy, A.H., Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer. methods partial differential equations, 25, 712-739, (2009) · Zbl 1170.65099
[8] Livermore, P.W.; Ierley, G.R., Quasi-\(L^p\) norm orthogonal Galerkin expansions in sums of Jacobi polynomials: orthogonal expansions, Numer. algorithms, 54, 533-569, (2010) · Zbl 1197.65027
[9] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Publications Mineola · Zbl 0987.65122
[10] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1989), Springer-Verlag New York
[11] Gheorghiu, C.I., ()
[12] Bialecki, B.; Fairweather, G.; Karageorghis, A., Matrix decomposition algorithms for elliptic boundary value problems: a survey, Numer. algor., 56, 253-295, (2011) · Zbl 1208.65036
[13] Doha, E.H.; Abd-Elhameed, W.M.; Bhrawy, A.H., Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations, Appl. math. model., 33, 1982-1996, (2009) · Zbl 1205.65224
[14] Doha, E.H.; Bhrawy, A.H.; Abd-Elhameed, W.M., Jacobi spectral Galerkin method for elliptic Neumann problems, Numer. algorithms, 50, 67-91, (2009) · Zbl 1169.65111
[15] Aghigh, K.; M-Jamei, M.; Dehghan, M., A survey on third and fourth kind of Chebyshev polynomials and their applications, Appl. math. comput., 199, 2-12, (2008) · Zbl 1134.33300
[16] Heinrichs, W., Spectral approximation of third-order problems, J. sci. comput., 14, 275-289, (1999) · Zbl 0953.65072
[17] J. Shen, Efficient Chebyshev-Legendre Galerkin methods for elliptic problems, in: A.V. Ilin, R. Scott (Eds.), Proc. ICOSAHOM’95, Houston J. Math., 1996, pp. 233-240.
[18] Shen, J., A new dual-petrov – galerkin method for third and higher odd-order differential equations: application to the KDV equations, SIAM J. numer. anal., 41, 1595-1619, (2003) · Zbl 1053.65085
[19] Ma, H.; Sun, W., A legendre – petrov – galerkin method and Chebyshev collocation method for the third-order differential equations, SIAM J. numer. anal., 38, 1425-1438, (2000) · Zbl 0986.65095
[20] Ma, H.; Sun, W., Optimal error estimates of the legendre – petrov – galerkin method for the korteweg – de Vries equation, SIAM J. numer. anal., 39, 1380-1394, (2001) · Zbl 1008.65070
[21] Livermore, P.W., Orthogonal Galerkin polynomials, J. comput. phys., 229, 2046-2060, (2010) · Zbl 1185.65138
[22] Fernandino, M.; Dorao, C.A.; Jakobsen, H.A., Jacobi Galerkin spectral method for cylindrical and spherical geometries, Chem. eng. sci., 62, 6777-6783, (2007)
[23] Huang, W.Z.; Sloan, D.M., The pseudospectral method for third-order differential equations, SIAM J. numer. anal., 29, 1626-1647, (1992) · Zbl 0764.65058
[24] Merryfield, W.J.; Shizgal, B., Properties of collocation third-derivative operators, J. comput. phys., 105, 182-185, (1993) · Zbl 0767.65074
[25] Luke, Y., The special functions and their approximations, vol. 2, (1969), Academic Press New York
[26] Doha, E.H., On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. phys. A: math. gen., 37, 657-675, (2004) · Zbl 1055.33007
[27] Guo, B.-Y., Spectral methods and their applications, (1998), World Scientific River Edge, NJ
[28] Guo, B.-Y.; Wang, L.-L., Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. approx. theory, 128, 1-41, (2004) · Zbl 1057.41003
[29] Li, H.; Wu, H.; Ma, H., The Legendre galerkin – chebyshev collocation method for Burgers-like equations, IMA J. numer. anal., 23, 109-124, (2003) · Zbl 1020.65072
[30] Don, W.S.; Gottlieb, D., The chebyshev – legendre method: implementing Legendre methods on Chebyshev points, SIAM J. numer. anal., 31, 1519-1534, (1994) · Zbl 0815.65106
[31] Wu, H.; Ma, H.; Li, H., Optimal error estimates of the chebyshev – legendre spectral method for solving the generalized Burgers equation, SIAM J. numer. anal., 41, 659-672, (2003) · Zbl 1050.65083
[32] Alpert, B.K.; Rokhlin, V., A fast algorithm for the evaluation of Legendre expansions, SIAM J. sci. statist. comput., 12, 158-179, (1991) · Zbl 0726.65018
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.