Abstract
Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving 2 nth-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 2 nth-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.
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