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

• Published in: Applied mathematical modelling Vol. 33; no. 4; pp. 1982 - 1996
• Main Authors: Doha, E.H., Abd-Elhameed, W.M., Bhrawy, A.H.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier 01.04.2009
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Partial differential equations; Sciences and techniques of general use; Spectral method; 2nth-order equations; Rounding error; Poisson equation; Inversion; Poisson's and biharmonic equations; Modeling; Ultraspherical polynomials; Nonhomogeneous Dirichlet conditions; Spectral-Galerkin method; Biharmonic equation; Dirichlet problem; Galerkin method
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2008.05.005

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.