We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on $[-1,1]$ whereas the latter have global support. The global approximation space can contain different affine transformations of the basis, mapping $[-1,1]$ to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving $K$ independent sparse linear systems of size $\mathcal{O}(n)\times \mathcal{O}(n)$, with $\mathcal{O}(n)$ nonzero entries, where $K$ is the number of different intervals and $n$ is the highest polynomial degree contained in the sum space. This results in an $\mathcal{O}(n)$ complexity solve. Applications to fractional heat and wave equations are considered.

翻译：我们针对定义在$\mathbb{R}$上的一类一维分数阶微分方程，提出了一种稀疏谱方法。这类方程可包含平方根拉普拉斯项、希尔伯特项、导数项及恒等项。该数值方法采用基于第二类加权切比雪夫多项式及其希尔伯特变换构成的基函数。前者支集位于$[-1,1]$区间内，而后者具有全局支集。全局逼近空间可包含该基函数的不同仿射变换，将$[-1,1]$映射至其他区间。值得注意的是，由此导出的线性系统不仅具有稀疏性，而且算子在不同仿射变换之间呈现解耦特性。因此，求解过程简化为求解$K$个独立的稀疏线性系统，每个系统规模为$\mathcal{O}(n)\times \mathcal{O}(n)$，非零元数量为$\mathcal{O}(n)$，其中$K$为不同区间个数，$n$为和空间中所含多项式最高次数。由此实现$\mathcal{O}(n)$复杂度的求解。本文还考虑了该方法在分数阶热传导方程与波动方程中的应用。