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.)
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[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
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