1 research outputs found
Generalised Hermite functions and their applications in Spectral Approximations
In 1939, G. Szeg\"{o} first introduced a family of generalised Hermite
polynomials (GHPs) as a generalisation of usual Hermite polynomials, which are
orthogonal with respect to the weight function |x|^{2\mu} \e^{-x^2},\mu>-\frac
12 on the whole line. Since then, there have been a few works on the study of
their properties, but no any on their applications to numerical solutions of
partial differential equations (PDEs). The main purposes of this paper are
twofold. The first is to construct the generalised Hermite polynomials and
generalised Hermite functions (GHFs) in arbitrary dimensions, which are
orthogonal with respect to |\bx|^{2\mu} \e^{-|\bx|^2} and |\bx |^{2\mu} in
respectively. We then define a family of adjoint generalised
Hermite functions (A-GHFs) upon GHFs, which has two appealing properties: (i)
the Fourier transform maps A-GHF to the corresponding GHF; and (ii) A-GHFs are
orthogonal with respect to the inner product
associated with the integral fractional Laplacian. The second purpose is to
explore their applications in spectral approximations of PDEs. As a remarkable
consequence of the fractional Sobolev-type orthogonality, the spectral-Galerkin
method using A-GHFs as basis functions leads to an identity stiffness matrix
for the integral fractional Laplacian operator which is known to
be notoriously difficult and expensive to discretise. ....Comment: 29 pages, 18 figure