Sinc-Galerkin Method Research Articles

The Sinc-Galerkin Method for Fourth-Order Differential Equations

The Sinc-Galerkin method originally proposed by Stenger is extended to handle fourth-order ordinary differential equations. The exponential convergence rate of the method, $O(e^{ - \kappa \sqrt M } )$ is carefully developed and the special features of the discrete system are described. Spectral properties and conditioning of the associated matrices are given. The appropriate choice of weight function in the Galerkin inner product is discussed with primary emphasis given to choices that are best suited to fourth-order partial differential equations. Numerical results are included to help illustrate the parameter selections made and confirm the efficiency and accuracy of the method.

The Sinc method in multiple space dimensions: Model problems

The Sinc-Galerkin method is applied to the canonical forms of second order partial differential equations in multiple space dimensions. For time dependent problems the scheme is fully Galerkin; i.e., the domain of the basis elements includes time. Hence the approximate solution for each equation is specified by solving the associated linear system. Notation is developed which facilitates description of both the linear systems and the algorithms used to solve them. Alternative algorithms are offered dependent on a scalar versus vector computing environment. Numerical results for the test problems presented sustain the exponential convergence rate of the method even in the presence of singularities.

Sinc-type approximations in H1-norm with applications to boundary value problems

Sinc functions are used for approximation on a general arc Γ in the complex plane. It is proved that for certain functions analytic in a domain containing the arc Γ, the error of an N-term approximation converges to zero, in the Sobolev space H 1(Γ), at the rate O(e −γ N 1 2 ), as N → ∞. Special attention is given to the important case of Γ = ( a, b), where a and b are finite real numbers. An application is also considered of the approximate result to the numerical solution of two-point boundary value problems. It is shown that the classical Sinc—Galerkin method with N basis functions has O(e −γ N 1 2 ) convergence rate in the H 1-norm.

Symmetrization of the Sinc-Galerkin Method with Block Techniques for Elliptic Equations

Description des techniques de blocs efficaces pour un systeme discret symetrique lie a la symetrisation de la methode de Galerkine-Sinc. Resultats numeriques

Symmetrization of the Sinc-Galerkin Method for Boundary Value Problems

On procede a la symetrisation de la methode de Galerkin en modifiant la fonction-poids pour une classe etendue de problemes aux valeurs limites

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 ( - \kappa \sqrt 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 ( - \kappa \sqrt n ) convergence rate.