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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.

65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE