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New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods. (English) Zbl 1286.65093
Summary: This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made.

##### MSC:
 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 41A10 Approximation by polynomials
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##### References:
 [1] Akram, Ghazala; Siddiqi, Shahid S., Solution of sixth order boundary value problems using non-polynomial spline technique, Appl. Math. Comput., 181, 708-720, (2006) · Zbl 1155.65361 [2] Alonso III, N.; Bowers, K. L., An alternating-direction sinc-Galerkin method for elliptic problems, J. Complexity, 25, 237-252, (2009) · Zbl 1166.65056 [3] Baldwin, P., Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global-phase integral methods, Philos. Trans. R. Soc. Lond. A, 322, 281-305, (1987) · Zbl 0625.76043 [4] Boyd, J. P., Chebyshev and Fourier spectral methods, (2001), Dover Mineola · Zbl 0994.65128 [5] 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 [6] Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods in fluid dynamics, (1989), Springer-Verlag New York · Zbl 0658.76001 [7] 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, 548-571, (2002) · Zbl 1020.65088 [8] Doha, E. H.; Abd-Elhameed, W. M., Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of ($$2 n + 1$$)th-order linear differential equations, Math. Comput. Simul., 79, 3221-3242, (2009) · Zbl 1169.65326 [9] E.H. Doha, W.M. Abd-Elhameed, On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds, B. Malays. Math. Sci. Soc., accepted for publication. · Zbl 1295.42012 [10] E.H. Doha, W.M. Abd-Elhameed, M.A. Bassuony, On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds , submitted for publication. · Zbl 1340.42063 [11] Doha, E. H.; Abd-Elhameed, W. M.; Bhrawy, A. H., Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of $$2 n$$th-order linear differential equations, Appl. Math. Modell., 33, 1982-1996, (2009) · Zbl 1205.65224 [12] 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 [13] 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 [14] Doha, E. H.; Bhrawy, A. H.; Abd-Elhameed, W. M., Jacobi spectral Galerkin method for elliptic neumman problems, Numer. Algor., 50, 67-91, (2009) · Zbl 1169.65111 [15] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations, Math. Comput. Modell., 53, 1820-1832, (2011) · Zbl 1219.65077 [16] 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 [17] Fox, L.; Parker, I. B., Chebyshev polynomials in numerical analysis, (1972), OUP Oxford [18] Islam, Siraj-Ul; Haq, Sirajul; Ali, Javid, Numerical solution of special 12th-order boundary value problems using differential transform method, Commun. Nonlinear Sci. Numer. Simul., 14, 1132-1138, (2009) · Zbl 1156.65090 [19] Gautschi, W., On mean convergence of extended Lagrange interpolation, J. Comput. Appl. Math., 43, 19-35, (1992) · Zbl 0761.41003 [20] C.I. Gheorghiu, Spectral Methods for Differential Problems, T. Popoviciu Institute of Numerical Analysis, Cluj-Napoca, Romania, 2007. [21] Gottlieb, D.; Orszag, S. A., Numerical analysis of spectral methods: theory and applications, (1977), SIAM Philadelphia · Zbl 0412.65058 [22] Graham, A., Kronecker product and matrix calculus with applications, (1981), Ellis Horwood Ltd. London [23] Julien, K.; Watson, M., Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods, J. Comput. Phys., 228, 1480-1503, (2009) · Zbl 1166.65325 [24] Lamnii, Abdelleh; Mraoui, Hamid; Sbibih, Driss; Tijini, Ahmed; Zidna, Ahmed, Spline collocation method for solving linear sixth-order boundary-value problems, Int. J. Comput. Math., 85, 1673-1684, (2008) · Zbl 1159.65079 [25] Lund, J., Symmetrization of the sinc-Galerkin method for boundary-value problems, Math. Comput., 47, 176, 571-588, (1986) · Zbl 0629.65085 [26] Mason, J. C., Chebyshev polynomial approximation for the L-membrane eigenvalue problem, SIAM J. Appl. Math., 15, 171-186, (1967) · Zbl 0149.36903 [27] Mason, J. C., Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms, J. Comput. Appl. Math., 49, 169-178, (1993) · Zbl 0793.33010 [28] Mason, J. C.; Handscomb, D. C., Chebyshev polynomials, (2003), Chapman and Hall New York, NY, CRC, Boca Raton · Zbl 1015.33001 [29] Notaris, S. E., Interpolatory quadrature formulae with Chebyshev abscissae of the third or fourth kind, J. Comput. Appl. Math., 81, 83-99, (1997) · Zbl 0881.65009 [30] Rashidinia, J.; Jalilian, R.; Farajeyan, K., Spline approximate solution of eighth-order boundary-value problems, Int. J. Comput. Math., 86, 1319-1333, (2009) · Zbl 1169.65327 [31] Twizell, E.; Boutayeb, A., Numerical methods for the solution of special and general sixthorder boundary value problems with applications to benard layer eigenvalue problems, Proc. R. Soc. Lond. A, 431, 433-450, (1990) · Zbl 0722.65042 [32] Zhi, Shi; Yong-yan, Cao, A spectral collocation method based on Haar wavelets for Poisson equations and biharmonic equations, Math. Comput. Modell., 54, 2858-2868, (2011) · Zbl 1235.65160
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