Müntz ball polynomials and Müntz spectral-Galerkin methods for singular eigenvalue problems

Abstract

In this paper, we introduce a new family of orthogonal systems, termed as the Müntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $\Vert x{\Vert }^{2\theta +2\mu -2}(1-\Vert x{\Vert }^{2\theta }{)}^{\alpha }$ with the parameters $\alpha >-1,\mu >-1/2$ and $\theta >0$ in the $d$-dimensional unit ball $x\in {\mathbb{B}}^d=\{x\in {\mathbb{R}}^d:r=\Vert x\Vert \le 1\}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schrödinger's operators with fractional potentials. We demonstrate that the use of such non-standard basis functions can not only tailor to the singularity of the solutions but also lead to sparse linear systems which can be solved efficiently.

22 pages, 7 figures