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Efficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of $\left(2n+1\right)$th-order linear differential equations. (English) Zbl 1236.65087
Summary: Some efficient and accurate algorithms based on the Legendre dual-Petrov Galerkin method are developed and implemented for solving $\left(2n+1\right)$th-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. The key idea to the efficiency of the algorithms is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The method leads to linear systems with specially structured matrices that can be efficiently inverted. Numerical results are presented to demonstrate the efficiency of the proposed algorithms.
##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 94B05 General theory of linear codes 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 65M70 Spectral, collocation and related methods (IVP of PDE) 65N35 Spectral, collocation and related methods (BVP of PDE) 35C10 Series solutions of PDE 42C10 Fourier series in special orthogonal functions