Spectral and pseudospectral methods for unbounded domains

Abstract

g 1 (ψ 1 (x, t), ψ 2 (x, t)) = i(|ψ 2 (x, t)| 2 − |ψ 1 (x, t)| 2)ψ 1 (x, t), g 2 (ψ 1 (x, t), ψ 2 (x, t)) = i(|ψ 1 (x, t)| 2 − |ψ 2 (x, t)| 2)ψ 2 (x, t).

Space–time spectral collocation method for Klein–Gordon equation

“…In general, one uses the spectral or pseudospectral method of Laguerre (or Hermite) for the problems defined on the half-line (or the whole line). [20][21][22][23][24][25] While the pseudospectral method is more preferable in actual computation, since it is easier to be implemented and saves work to deal with nonlinear terms. 22 Spectral collocation methods are efficient techniques for solving nonlinear equations accurately.…”

“…[20][21][22][23][24][25] While the pseudospectral method is more preferable in actual computation, since it is easier to be implemented and saves work to deal with nonlinear terms. 22 Spectral collocation methods are efficient techniques for solving nonlinear equations accurately. 26 In this paper, we will develop a Laguerre-Legendre spectral collection method to construct a scheme to calculate the numerical solution of NLKG equation (1) defined on the half-line with initial value conditions:…”

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

“…Recently, Guo [15,16], Shen and Wang [35] reviewed some results on the spectral methods for problems defined on unbounded domains. In general, there are two types of techniques (cf.…”

“…For the partial differential equations of non-standard types, we prefer to use composite orthogonal approximations and interpolation methods of the Laguerre (Hermite) polynomials/functions and Jacobi polynomials with the domain decomposition; see [8,16,26,38] and the references therein.…”

A Domain Decomposition Based Spectral Collocation Method for Lane-Emden Equations

“…We mention in passing that the domain decomposition methods are a very effective tool to solve a variety of differential equations whose coefficient functions are singular or degenerate around the boundary of the solution domain, since we can use different suitable and efficient spectral methods in different domains. We refer to [8] for details along this line. In order to derive the error estimates of the proposed spectral collocation method, we require first to establish error estimates for the multi-step LGR collocation method based on the formulation (1.2) and the LGR collocation method for the method (1.3).…”