Efficient solutions of multidimensional sixth-order boundary value problems using symmetric generalized Jacobi-Galerkin method. (English) Zbl 1246.65121

Summary: This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities.
These algorithms are extensions to some of the algorithms proposed by the authors [SIAM J. Sci. Comput. 24, No. 2, 548–571 (2002; Zbl 1020.65088)] and A. H. Bhrawy and the first author [Appl. Numer. Math. 58, No. 8, 1224–1244 (2008; Zbl 1152.65112)] for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations,” Abstract and Applied Analysis, Article ID 947230, 21 pages, 2011. · Zbl 1216.65086 · doi:10.1155/2011/947230
[2] A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “Spectral shifted Jacobi tau and collocation methods for solving fifth-order boundary value problems,” Abstract and Applied Analysis, vol. 2011, Article ID 823273, 14 pages, 2011. · Zbl 1221.65171 · doi:10.1155/2011/823273
[3] G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publication, 1985. · Zbl 0023.21505
[4] G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71, Cambridge University Press, Cambridge, UK, 1999. · Zbl 0920.33001
[5] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Mineola, NY, 2nd edition, 2001. · Zbl 0994.65128
[6] D. Funaro, Polynomial approximation of differential equations, vol. 8 of Lecture Notes in Physics, Springer, Berlin, Germany, 1992. · Zbl 0774.41010
[7] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1989. · Zbl 0717.76004
[8] A. Boutayeb and E. H. Twizell, “Numerical methods for the solution of special sixthorder boundary-value problems,” International Journal of Computer Mathematics, vol. 45, no. 3, pp. 207-223, 1992. · Zbl 0773.65055 · doi:10.1080/00207169208804130
[9] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford, UK, 1961. · Zbl 0142.44103
[10] R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific, Singapore, 1986. · Zbl 0681.76121
[11] A. H. Bhrawy, “Legendre-Galerkin method for sixth-order boundary value problems,” Journal of the Egyptian Mathematical Society, vol. 17, no. 2, pp. 173-188, 2009. · Zbl 1190.65177
[12] A. Lamnii, H. Mraoui, D. Sbibih, A. Tijini, and A. Zidna, “Spline collocation method for solving linear sixth-order boundary-value problems,” International Journal of Computer Mathematics, vol. 85, no. 11, pp. 1673-1684, 2008. · Zbl 1159.65079 · doi:10.1080/00207160701543392
[13] S. S. Siddiqi and E. H. Twizell, “Spline solutions of linear sixth-order boundary-value problems,” International Journal of Computer Mathematics, vol. 60, no. 3-4, pp. 295-304, 1996. · Zbl 1001.65523 · doi:10.1080/00207169608804493
[14] M. El-Gamel, J. R. Cannon, and A. I. Zayed, “Sinc-Galerkin method for solving linear sixth-order boundary-value problems,” Mathematics of Computation, vol. 73, no. 247, pp. 1325-1343, 2004. · Zbl 1054.65085 · doi:10.1090/S0025-5718-03-01587-4
[15] B.-Y. Guo, J. Shen, and L.-L. Wang, “Optimal spectral-Galerkin methods using generalized Jacobi polynomials,” Journal of Scientific Computing, vol. 27, no. 1-3, pp. 305-322, 2006. · Zbl 1102.76047 · doi:10.1007/s10915-005-9055-7
[16] E. H. Doha and W. M. Abd-Elhameed, “Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials,” SIAM Journal on Scientific Computing, vol. 24, no. 2, pp. 548-571, 2002. · Zbl 1020.65088 · doi:10.1137/S1064827500378933
[17] E. H. Doha and W. M. Abd-Elhameed, “Efficient spectral ultraspherical-dual-Petrov-Galerkin algorithms for the direct solution of (2n+1) th-order linear differential equations,” Mathematics and Computers in Simulation, vol. 79, no. 11, pp. 3221-3242, 2009. · Zbl 1169.65326 · doi:10.1016/j.matcom.2009.03.011
[18] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224-1244, 2008. · Zbl 1152.65112 · doi:10.1016/j.apnum.2007.07.001
[19] E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations,” Applied Mathematical Modelling, vol. 33, no. 4, pp. 1982-1996, 2009. · Zbl 1205.65224 · doi:10.1016/j.apm.2008.05.005
[20] E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials,” Quaestiones Mathematicae, vol. 218, no. 15, pp. 7727-7740, 2012. · Zbl 1242.65148
[21] M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematical Series, National Bureau of Standards, New York, NY, USA, 1970.
[22] E. H. Doha, “The coefficients of differentiated expansions and derivatives of ultraspherical polynomials,” Computers and Mathematics with Applications, vol. 21, no. 2-3, pp. 115-122, 1991. · Zbl 0723.33008 · doi:10.1016/0898-1221(91)90089-M
[23] E. H. Doha, “On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations,” Journal of Computational and Applied Mathematics, vol. 139, no. 2, pp. 275-298, 2002. · Zbl 0991.33003 · doi:10.1016/S0377-0427(01)00420-4
[24] M. Schatzman and J. Taylor, Numerical Analysis: A Mathematical Introduction, Clarendon Press, Oxford, UK, 2002. · Zbl 1019.65003
[25] E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics A, vol. 37, no. 3, pp. 657-675, 2004. · Zbl 1055.33007 · doi:10.1088/0305-4470/37/3/010
[26] A. Graham, Kronecker Product and Matrix Calculus With Applications, Ellis Horwood, London, UK, 1981. · Zbl 0497.26005
[27] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001
[28] G. Akram and S. S. Siddiqi, “Solution of sixth order boundary value problems using non-polynomial spline technique,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 708-720, 2006. · Zbl 1155.65361 · doi:10.1016/j.amc.2006.01.053
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