Müntz ball polynomials and Müntz spectral-Galerkin methods for singular eigenvalue problems - Yang, Xiu (2023)
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∈ Bd= { x∈ Rd: 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.