A frame approach for equations involving the fractional Laplacian
File(s)
Accepted version
Author(s)
Papadopoulos, Ioannis
Gutleb, Timon
Carrillo, Jose
Olver, Sheehan
Type
Journal Article
Abstract
Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical
orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, s ∈ (0, 1), on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to Rd , d ∈ {1, 2}. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a 6th-order Runge–Kutta time discretization), a fractional heat equation with a time-dependent exponent s(t), and a two-dimensional problem, observing spectral convergence in the
spatial dimension for sufficiently smooth data.
orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, s ∈ (0, 1), on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to Rd , d ∈ {1, 2}. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a 6th-order Runge–Kutta time discretization), a fractional heat equation with a time-dependent exponent s(t), and a two-dimensional problem, observing spectral convergence in the
spatial dimension for sufficiently smooth data.
Date Issued
2025-11-08
Date Acceptance
2025-07-22
Citation
IMA Journal of Numerical Analysis, 2025
ISSN
0272-4979
Publisher
Oxford University Press
Journal / Book Title
IMA Journal of Numerical Analysis
Copyright Statement
© The Author(s) 2025. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published online
Date Publish Online
2025-11-08