  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.
