Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations

Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2nth-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional 2nth-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.