Approximation of some diffusion evolution equations in unbounded domains by Hermite functions

Abstract

Spectral and pseudospectral approximations of the heat equation are analyzed. The solution is represented in a suitable basis constructed with Hermite polynomials. Stability and convergence estimates are given and numerical tests are discussed.

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

“…If α = 1 2 , then the H n (v) becomes the Hermite function as discussed in Funaro and Kavian [17]. Theorem 2.1 generalizes the corresponding results in [17], while other results in this section are new, which make the use of the method of Funaro and Kavian possible for more general problems.…”

“…In many problems arising in quantum mechanics and statistical physics, the solutions decay exponentially as |v| → ∞. In this case, it is reasonable to take the basis functions as those used in Funaro and Kavian [17], or as in Tang et al [40].…”

“…Some earlier works on the convergence analysis of spectral methods in unbounded domains have been given by Funaro and Kavian [17], Guo [23] (on Hermite spectral approximations); by Mavriplis [32], Shen [37] (on Laguerre approximations); by Boyd [6], Grosch and Orszag [20] (on rational polynomial approximations). Although Hermite-spectral approximations have been used successfully in approximating the solutions to the Fokker-Planck equations (and also the Vlasov equations), there has been little convergence analysis for these numerical schemes.…”

Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel

“…[1,2,6,9,11,12]), considerable progress has been made recently in spectral methods for unbounded domains. Among these methods, a direct and commonly used approach is based on certain orthogonal approximations on infinite intervals, i.e., the Hermite and Laguerre spectral methods ( see, e.g., [5,10,13,16,18,23,24,28,32]). Some authors also developed composite Laguerre-Legendre spectral methods for the half line and mixed Laguerre-Legendre spectral methods for an infinite strip; see [9,15,21,31].…”

Hermite spectral and pseudospectral methods for nonlinear partial differential equation in multiple dimensions

“…Gottlieb and Orszag [1], Maday, Pernaud-Thomas and Vandeven [2], Coulaud, Funaro and Kavian [3], Funaro [4], and Guo and Shen [5] developed the Laguerre spectral method. While Funaro and Kavian [6] provided some numerical algorithms by using Hermite functions. Furthermore, Guo [7] established some approximation results on the Hermite polynomial approximation with applications to partial differential equations.…”