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A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations. (English) Zbl 1219.65077
Summary: This paper analyzes a method for solving the third- and fifth-order differential equations with constant coefficients using a Jacobi dual-Petrov-Galerkin method, which is more reasonable than the standard Galerkin one. The spatial approximation is based on Jacobi polynomials ${P}_{n}^{\alpha ,\beta }$ with $\alpha ,\beta \in \left(-1,\infty$) and $n$ is the polynomial degree. By choosing appropriate base functions, the resulting system is sparse and the method can be implemented efficiently. A Jacobi-Jacobi dual-Petrov-Galerkin method for the differential equations with variable coefficients is developed. This method is based on the Petrov-Galerkin variational form of one Jacobi polynomial class, but the variable coefficients and the right-hand terms are treated by using the Gauss-Lobatto quadrature form of another Jacobi class. Numerical results illustrate the theory and constitute a convincing argument for the feasibility of the proposed numerical methods.
##### MSC:
 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 33C45 Orthogonal polynomials and functions of hypergeometric type