A frame approach for equations involving the fractional Laplacian.
CoRR(2023)
摘要
Exceptionally elegant formulae exist for the fractional Laplacian operator
applied to weighted classical orthogonal polynomials. We utilize these results
to construct a spectral method, based on frame properties, for solving
equations involving the fractional Laplacian of any power, $s \in (0,1)$, 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 $\mathbb{R}^d$,
$d \in \{1,2\}$. We examine the frame properties of our approximation space
and, under standard frame conditions, 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 $6^\text{th}$-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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