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Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in(More)
A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are con-vergent on a semi-infinite interval with spectral accuracy. An efficient and(More)
Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces are investigated. Some results on orthogonal projections and interpolations are established. Explicit expressions describing the dependence of approximation results on the parameters of Jacobi polynomials are given. These results serve as an important tool in the analysis of numerous(More)
We extend the definition of the classical Jacobi polynomials withindexes α, β > −1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this(More)
We propose a new spectral viscosity (SV) scheme for the accurate solution of non-linear conservation laws. It is proved that the SV solution converges to the unique entropy solution under appropriate reasonable conditions. The proposed SV scheme is implemented directly on high modes of the computed solution. This should be compared with the original(More)
We consider in this paper spectral and pseudospectral approximations using Hermite functions for PDEs on the whole line. We first develop some basic approximation results associated with the projections and interpolations in the spaces spanned by Hermite functions. These results play important roles in the analysis of the related spectral and pseudospectral(More)
In this paper, error estimates for generalized Laguerre–Gauss-type interpolations are derived in nonuniformly weighted Sobolev spaces weighted with ω α,β (x) = x α e −βx , α > −1, β > 0. Generalized Laguerre pseudospectral methods are analyzed and implemented. Two model problems are considered. The proposed schemes keep spectral accuracy and, with suitable(More)
We study a Fourier-spectral method for a dissipative system modeling the flow of liquid crystals. We first prove its convergence in a suitable sense and establish the existence of a global weak solution of the original problem and its uniqueness in the two dimensional case. Then we derive error estimates which exhibit the spectral accuracy of the(More)