A frame approach for equations involving the fractional Laplacian
Authors
- Papadopoulos, Ioannis
- Gutleb, Timon
- Carrillo, José
- Olver, Sheehan
2020 Mathematics Subject Classification
- 26A33 33C45 33C50 65D05
Keywords
- Frame, fractional Laplacian, fractional calculus
DOI
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.
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