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.
翻译:在本文中,我们引入了一个新的正方形系统(MBPPs),称为M\"{u}ntz ball 多元球(MBPPs),在重量函数方面是正方形的:$<unk> 2\\theta+2\mu-2}(1-<unk> x<unk> 2\\theta}}}<unk> alpha}$,参数为$\alpha>-1,\mu>-1.2美元和$\theta>0美元。我们证明,使用这种非标准功能不仅能够适应解决方案的单一性,而且能够导致可有效解决的稀薄线性系统。</s>