Efficient spectral-Petrov–Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials

This article analyzes some algorithms for solving numerically the integrated forms of third- and fifth-order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. Two new families of general parameters generalized Jacobi polynomials are introduced and used for this purpose. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented.