title: New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov-Galerkin method.
abstract: Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient.
authors.affiliation: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
authors.forename: E H
authors.initials: EH
authors.lastname: Doha
authors.affiliation: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt ; Department of Mathematics, Faculty of Science, King Abdulaziz University, Saudi Arabia.
authors.forename: W M
authors.initials: WM
authors.lastname: Abd-Elhameed
authors.affiliation: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
authors.forename: Y H
authors.initials: YH
authors.lastname: Youssri
keywords: Convergence analysis; Dual-Petrov–Galerkin method; Generalized Jacobi polynomials; Nonhomogeneous Dirichlet conditions