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

2023/03/09 by Yang, Xiu, Wang, Li-Lian, Li, Huiyuan +1
#33C45 #33C55 #35Q40 #47A75 #65N25 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2303.05020

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: ‖x‖^2θ+2μ-2 (1-‖x‖^2θ)^α with the parameters α>-1, μ>- 1/2 and θ>0 in the d-dimensional unit ball x∈ \mathbb B^d=\x∈ℝ^d: r=‖x‖≤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.

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