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 M\"{u}ntz ball polynomials (MBPs), which are orthogonal with respect to the weight function: $|x|^{2\theta+2\mu-2} (1-|x|^{2\theta})^{\alpha}$ with the parameters $\alpha>-1, \mu>- 1/2$ and $\theta>0$ in the $d$-dimensional unit ball $x\in {\mathbb B}^d=\big{x\in\mathbb{R}^d: r=|x|\leq1\big}$. We then develop efficient and spectrally accurate MBP spectral-Galerkin methods for singular eigenvalue problems including degenerating elliptic problems with perturbed ellipticity and Schr\"odinger'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.