Hermite Spectral Collocation Methods for Fractional PDEs in Unbounded Domains

Abstract

This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite points. In this work, two Hermite-type functions are employed to serve as basis functions. Our main task is to find corresponding differentiation matrices which are computed recursively. Two important issues relevant to condition numbers and scaling factors will be discussed. Applications of the spectral collocation methods to multi-term fractional PDEs are also presented. Several numerical examples are carried out to demonstrate the effectiveness of the proposed methods.

Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

“…Modified rational spectral-collocation methods. With the formulas in Theorem 3.4 at our disposal, we can directly generate the spectral fractional differentiation matrices and develop the direct collocation methods, similar to the Hermite collocation methods in [39]. However, it seems nontrivial and largely open to analyse Left: Hermite function approach in [28] with scaling factor 1/0.7.…”

“…Notice that the collocation method is more practical for problems with variable coefficients and nonlinear problems. Also, we shall carry out comparisons with the Hermite collocation method in [39]. Here we set J = 4 and α 1 = 0, α 2 = 0.5 α 3 = 1.5, α 4 = 2,…”

“…For example, the key to the Hermite spectral method in [28] is the use of the attractive property that the Hermtie functions are the eigenfunctions of the Fourier transform, so the algorithm is largely implemented in the frequency ξ-space. In contrast, some analytically perspicuous formulas of (−∆) α/2 on the Hermite functions were derived in [39], which led themselves to the construction of efficient collocation algorithms in the physical x-space. In the spirit of [39], we search for the analytic formulas for computing the fractional Laplacian of the rational basis functions -the modified mapped Gegenbauer functions (MMGFs), orthogonal with respect to a uniform weight.…”

“…In contrast, some analytically perspicuous formulas of (−∆) α/2 on the Hermite functions were derived in [39], which led themselves to the construction of efficient collocation algorithms in the physical x-space. In the spirit of [39], we search for the analytic formulas for computing the fractional Laplacian of the rational basis functions -the modified mapped Gegenbauer functions (MMGFs), orthogonal with respect to a uniform weight. Although the formulas (see Theorem 3.4) are not as compact as those for the Hermite functions, we can accurately compute the fractional Laplacian of the rational basis up to the degree ∼ 10 3 by using e.g., Maple or Mathematica.…”

“…In this section, we consider the spectral-Galerkin approximation to a model equation, and conduct the error analysis. We also present some numerical results to show our proposed method outperforms the Hermite approximations in [28,39].…”