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

03/09/2023

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by   Xiu Yang, et al.

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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∈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.