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arXiv · 2303.05020

M\"untz ball polynomials and M\"untz spectral-Galerkin methods for singular eigenvalue problems

Xiu Yang·Li-Lian Wang·Huiyuan Li·Changtao Sheng
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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.

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Xiu Yang, Li-Lian Wang, Huiyuan Li, Changtao Sheng. 2023-03-09. M\"untz ball polynomials and M\"untz spectral-Galerkin methods for singular eigenvalue problems. https://arxiv.org/abs/2303.05020

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