# zbMATH — the first resource for mathematics

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations. (English) Zbl 1205.65224
Summary: Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving $$2n$$th-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional $$2n$$th-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.

##### MSC:
 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Boutayeb, A.; Twizell, E., Numerical methods for the solution of special sixth-order boundary value problems, Int. J. comput. math., 45, 207-233, (1992) · Zbl 0773.65055 [2] Twizell, E.; Boutayeb, A., Numerical methods for the solution of special and general sixth-order boundary value problems with applications to benard layer eigenvalue problems, Proc. roy. soc. London A., 431, 433-450, (1990) · Zbl 0722.65042 [3] Baldwin, P., Asymptotic estimates of the eigenvalues of a sixth order boundary value problem obtained by using global phase-integral methods, Phil. trans. roy. soc. London A, 322, 281-305, (1987) · Zbl 0625.76043 [4] El-Gamel, M.; Cannon, J.R.; Zayed, A.I., Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. comput., 73, 1325-1343, (2003) · Zbl 1054.65085 [5] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York-Heidelberg-Berlin · Zbl 0658.76001 [6] Coutsias, E.A.; Hagstrom, T.; Torres, D., An efficient spectral method for ordinary differential equations with rational function coefficients, Math. comput., 65, 611-635, (1996) · Zbl 0846.65037 [7] Doha, E.H., An accurate double Chebyshev spectral approximations for poisson’s equation, Ann. univ. sci. Budapest. sect. comp., 10, 243-275, (1990) · Zbl 0717.65090 [8] Doha, E.H., An accurate solution of parabolic equations by expansion in ultraspherical polynomials, J. comp. math. appl., 21, 4, 75-88, (1990) · Zbl 0706.65089 [9] Doha, E.H., The ultraspherical coefficients of the moments of a general – order derivative of an infinitely differentiable function, J. comp. appl. math., 89, 53-72, (1998) · Zbl 0909.33007 [10] Doha, E.H., On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations, J. comp. appl. math., 139, 275-298, (2002) · Zbl 0991.33003 [11] Doha, E.H.; Abd-Elhameed, W.M., Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method, J. comput. appl. math., 181, 24-45, (2005) · Zbl 1071.65136 [12] Doha, E.H.; Al-Kholi, F.M.R., An efficient double Legendre spectral method for parabolic and elliptic partial differential equations, Int. J. comput. math., 78, 413-432, (2001) · Zbl 1004.65101 [13] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977. · Zbl 0412.65058 [14] Heinrichs, W., Improved condition number for spectral methods, Math. comput., 53, 103-119, (1989) · Zbl 0676.65115 [15] Heinrichs, W., Spectral methods with sparse matrices, Numer. math., 56, 25-41, (1989) · Zbl 0661.65120 [16] Karageorghis, A., Chebyshev spectral methods for solving two-point boundary value problem arising in the heat transfer, J. comput. methods appl. mech. eng., 70, 103-121, (1988) · Zbl 0636.65083 [17] Doha, E.H.; Abd-Elhameed, W.M., Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. sci. comput., 24, 2, 548-571, (2002) · Zbl 1020.65088 [18] E.H. Doha, W.M. Abd-Elhameed, Efficient spectral ultraspherical-Galerkin approximations of one-dimensional fourth-order problems, in: E.H. Doha, M.A. Helal (Eds.), in: Proceedings of the international conference on mathematics: Trends and Developments, Egyptian Math. Soc., Cairo 28-31 Dec. 2002, vol. III, Fluid Mechanics and Numerical Analysis, (2003), pp. 99-126. [19] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. algor., 42, 137-164, (2006) · Zbl 1103.65119 [20] 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 [21] 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 [22] E.H. Doha, Spectral solutions of differential and difference equations with polynomial coefficients by using classical orthogonal polynomials, J. Comput. Appl. Math., in press. [23] Shen, J., Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. sci. comput., 15, 1489-1505, (1994) · Zbl 0811.65097 [24] Shen, J., Efficient spectral-Galerkin method II: direct solvers of second-and fourth-order equations using Chebyshev polynomials, SIAM J. sci. comput., 16, 74-87, (1995) · Zbl 0840.65113 [25] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge [26] Doha, E.H., The coefficients of differentiated expansions and derivatives of ultraspherical polynomials, J. comp. math. appl., 19, 2-3, 115-122, (1991) · Zbl 0723.33008 [27] E.H. Doha, Fathalla A. Rihan, W.M. Abd-Elhameed, Efficient ultraspherical spectral-Galerkin algorithms for the approximation of fourth-order equations, submitted for publication. [28] 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 [29] Graham, A., Kronecker product and matrix calculus with applications, (1981), Ellis Horwood Ltd. London [30] Funaro, D., Polynomial approximation of differential equations, Lecturer notes in physics, (1992), Springer-Verlag Heidelberg, Berlin, New York [31] Luke, Y., The special functions and their approximations, vol. 1, (1969), Academic Press New York
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.