Symmetrization of the sinc-Galerkin method for boundary value problems

Abstract

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 <italic>n</italic> function evaluations, the method of [5] converges at the rate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis minus kappa StartRoot n EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>κ<!-- κ --></mml:mi> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp ( - \kappa \sqrt n )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <italic>k</italic> is independent of <italic>n</italic>. 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis minus kappa StartRoot n EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>κ<!-- κ --></mml:mi> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\exp ( - \kappa \sqrt n )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> convergence rate.