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：对于作用于加权经典正交多项式上的分数阶拉普拉斯算子，存在极为优雅的公式。我们利用这些结果，基于框架性质，构建了一种求解一维或二维无界域上任意幂次分数阶拉普拉斯算子方程（$s \in (0,1)$）的求解器。该数值方法将解展开为加权经典正交多项式及其在$\mathbb{R}^d$（$d \in \{1,2\}$）中特定延拓下的非加权对应项。我们考察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平稳方程的先验估计。此外，对于分数阶热方程的时间隐式欧拉离散化，我们证明该方法能达到预期的收敛阶。我们将求解器应用于众多算例，包括分数阶热方程（采用高达六阶龙格-库塔时间离散化）、含时变指数$s(t)$的分数阶热方程以及二维问题，并观察到对于足够光滑的数据，空间维度呈现出谱收敛性。