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.

翻译：针对作用于加权经典正交多项式的分数阶拉普拉斯算子，存在异常简洁的公式。我们利用这些结果，基于框架性质构建了一种谱方法，用于求解定义在一维或二维无界域上、任意幂次 $s \in (0,1)$ 的分数阶拉普拉斯方程。该数值方法将解表示为加权经典正交多项式及其无加权形式（通过特定延拓至 $\mathbb{R}^d$，$d \in \{1,2\}$）的展开式。我们检验了逼近空间的框架性质，并在标准框架条件下证明：当对分数阶热方程采用时间隐式Euler离散时，可获得预期的收敛阶数。我们将求解器应用于多个算例，包括分数阶热方程（采用高达6阶Runge-Kutta时间离散）、含时变指数 $s(t)$ 的分数阶热方程以及二维问题，对于足够光滑的数据观察到空间维度的谱收敛性。