Symmetrization of the sinc-Galerkin method for boundary value problems

The Sinc-Galerkin method developed in [5], when applied to the second-order selfadjoint boundary value problem, gives rise to a nonsymmetric coefficient matrix. The technique in [5] is based on weighting the Galerkin inner products in such a way that the method will handle boundary value problems with regular singular points. In particular, the method does an accurate job of handling problems with singular solutions (the first or a higher derivative of the solution is unbounded at one or both of the boundary points). Using n function evaluations, the method of [5] converges at the rate exp ⁡ ( − κ n ) exp ( - κ √ n ) , where k is independent of n . In this paper it is shown that, by changing the weight function used in the Galerkin inner products, the coefficient matrix can be made symmetric. This symmetric method is applicable to a slightly more restrictive set of boundary value problems than the method of [5], The present method, however, still handles a wide class of singular problems and also has the same exp ⁡ ( − κ n ) exp ( - κ √ n ) convergence rate.