## Efficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of $$(2n+1)$$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 $$(2n+1)$$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 Numerical solution of boundary value problems involving ordinary differential equations 94B05 Linear codes (general theory) 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35C10 Series solutions to PDEs 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)