A Compendium of Galerkin Orthogonal Polynomials


February 5, 2014
This online resource concerns the construction of a polynomial Galerkin basis {Ψn(x)}, each member of which satisfies two key properties (i) any given set of M homogeneous linear boundary conditions up to degree N - 1 and (ii) orthogonality to any other basis function. Although a brief overview of Galerkin methods — and the usage of this database — is provided below, far more detail regarding the theory and applications is contained in [2] and [3].

In the left hand frame of the page is listed a range of boundary conditions for (a) orthogonality in Cartesian coordinates, (b) orthogonality in polar geometries and (c) more general orthogonality relations involving derivatives. Common and physically motivated boundary conditions are given explicitly, followed by more general boundary conditions as far as the extent of computer algebra allows. Simply find the boundary condition of interest, and copy and paste the text defining the basis set. In the cases where α and β are unspecified, simply substitute your particular preference.

First published online 2009 at http://escholarship.org/uc/item/9vk1c6cm; updated Feb 2014.