We develop a sparse spectral method for a class of fractional differential equations, posed on \(\mathbb {R}\), in one dimension. These equations may 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 may 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.

我们为一类分数阶微分方程开发了一种稀疏谱方法，在一维上提出于 \(\mathbb {R}\)。这些方程可能包括 sqrt-Laplacian、Hilbert、导数和恒等项。该数值方法利用由第二类加权切比雪夫多项式及其希尔伯特变换组成的基础。前一个函数在 \([-1,1]\) 上受支持，而后者则具有全局支持。全局逼近空间可能包含基的不同仿射变换，将 \([-1,1]\) 映射到其他区间。值得注意的是，不仅诱导线性系统稀疏，而且算子在不同的仿射变换之间解耦。因此，求解简化为求解 K 个大小为 \(\mathcal {O}(n)\times \mathcal {O}(n)\) 的独立稀疏线性系统，其中 \(\mathcal {O}(n)\)非零项，其中 K 是不同区间的数量，n 是和空间中包含的最高多项式次数。这导致 \(\mathcal {O}(n)\) 复杂度求解。考虑了分数热方程和波动方程的应用。