论文标题
一种稀疏的光谱方法，用于单空间维度的分数微分方程
A sparse spectral method for fractional differential equations in one-spatial dimension
论文作者
论文摘要
我们为一类分数微分方程开发了一种稀疏的光谱方法，该方法在一个维度上在$ \ mathbb {r} $上提出。这些方程可以包括SQRT-Laplacian，Hilbert，衍生和身份术语。该数值方法利用了由第二种的加权Chebyshev多项式和希尔伯特转换组成的基础。前者的功能得到$ [-1,1] $的支持，而后者则得到全球支持。全局近似空间可以包含基础的不同仿射转换，将$ [-1,1] $映射到其他间隔。值得注意的是，不仅诱导的线性系统稀疏，而且运算符将不同的仿射转换分解。 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.这导致$ \ Mathcal {O}（n）$复杂性解决。考虑到分数热和波动方程的应用。
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.
