Nonsymmetric Generalized Jacobi Petrov-Galerkin Algorithms for Third- and Fifth-Order two Point Boundary Value Problems

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are used for solving third- and fifth-order two point boundary value problems subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The key 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 method leads to linear systems with specially structured matrices that can be efficiently inverted. The use of generalized Jacobi polynomials leads to simplified analysis, very efficient numerical algorithms. Numerical results are presented to demonstrate the efficiency of our proposed algorithms.