Generalized Hermite Spectral Method and its Applications to Problems in Unbounded Domains

Time-dependent Hermite–Galerkin spectral method and its applications

“…And it is as we expected that the time-dependent scaling factor yields better resolution than the constant one, i.e. the scheme in [26,17]. From the viewpoint of computational complexity, our scheme can be implemented straight-forward without any variable transformation.…”

“…This equivalence can be extended trivially to where β = 0, by mimicking the proof of Lemma 2.2-2.3 in [26] with x replaced by (x − β). From the definition of || · || P,r,α , it is clear to see that the larger r is, the smaller space H r,α is, and the smoother the functions are.…”

“…It is readily shown in [26] and [27] that when β = 0 the norm defined in (2.9) is equivalent to the following Sobolev-like norm…”

“…In this example, we shall investigate our scheme with the time-invariant scaling factor α = . The HSM with time-invariant scaling factor is essentially developed in [26,17] and can be viewed as a special case of our THGSM. The accuracy of the numerical results obtained by our THGSM and the time-invariant scheme are compared with some previous results, including the ones in [6], [18].…”

A class Hermite pseudospectral approximate with ω(x) ≡ 1 and application to reaction-diffusion equation

“…In [12], making use of the idea of [1,8], we present a Hermite function system with weight ω(x) ≡ 1 and established the spectral approximate theory for the whole line. The aim of this paper is to develop the Hermite pseudospectral method.…”