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 Pn(α,β) with α,β∈(-1,∞) 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. © 2011 Elsevier Ltd.
Doha, E. H., Bhrawy, A. H., & Hafez, R. M. (2011). A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations. Mathematical and Computer Modelling, 53(9–10), 1820–1832. https://doi.org/10.1016/j.mcm.2011.01.002