A frame approach for equations involving the fractional Laplacian

TL;DR

This paper develops a novel solver for fractional Laplacian equations on unbounded domains using frame properties of weighted orthogonal polynomials, achieving spectral convergence and handling time-dependent exponents.

Contribution

It introduces a frame-based spectral method for fractional Laplacian equations, providing convergence analysis and applying it to complex problems including time-dependent exponents.

Findings

Achieves spectral convergence for smooth data

Provides an a priori estimate for the stationary problem

Successfully applies to 1D and 2D fractional heat equations

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, $s \in (0,1)$, 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 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 $6^\text{th}$-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.