A Jacobi–Jacobi dual-Petrov–Galerkin method for third- and fifth-order differential equations

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,∞)α,β∈(−1,∞) and nn 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.