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.

翻译：加权经典正交多项式在分数拉普拉斯算子作用下存在异常优美的表达式。我们利用这些结果，基于框架性质构建了一种求解器，用于求解一维或二维无界域上任意幂次 $s \in (0,1)$ 的分数拉普拉斯方程。该数值方法将解表示为加权经典正交多项式及其未加权对应项的展开形式，并特别推广至 $\mathbb{R}^d$（$d \in \{1,2\}$）。我们研究了该函数族在解展开中的框架性质，并在标准框架条件下推导了稳态方程的先验估计。此外，我们证明了在分数热方程中采用隐式欧拉时间离散时能达到预期的收敛阶数。我们将求解器应用于多个算例，包括分数热方程（采用高达六阶龙格-库塔时间离散）、具有时变指数 $s(t)$ 的分数热方程以及二维问题，在数据充分光滑时观测到空间维度上的谱收敛性。