A Pseudospectral Method for the One-Dimensional Fractional Laplacian on $\mathbb R$

使用移位 Gegenbauer 伪谱方法对高阶 Caputo 分数阶导数的数值逼近：Bagley-Torvik 型两点边值问题案例研究

The Numerical Approximation of Caputo Fractional Derivative of Higher Orders Using A Shifted Gegenbauer Pseudospectral Method: Two-Point Boundary Value Problems of the Bagley Torvik Type Case Study

本研究提出一种移位 Gegenbauer 伪谱方法逼近任意正阶 Caputo 分数阶导数，通过引入分数阶 SG 积分矩阵实现高效计算，在求解 Bagley-Torvik 型两点边值问题时展现出优于现有方法的指数收敛精度与稳定性。

用于弱正则右侧Caputo终端问题的非标准分数阶Jacobi伪谱方法

A nonstandard fractional Jacobi pseudospectral method for weakly regular right-sided Caputo terminal problems

针对弱正则右侧Caputo终端问题，提出非标准分数阶Jacobi伪谱方法，可准确逼近终端附近正则性有限的解，精度优于传统谱方法，为相关系统提供高效计算工具。

求解具有变系数和周期解的含时一维分数阶偏微分方程的 Fourier-Gegenbauer 伪谱方法

Fourier-Gegenbauer Pseudospectral Method for Solving Time-Dependent One-Dimensional Fractional Partial Differential Equations with Variable Coefficients and Periodic Solutions

针对带变系数与周期解的含时一维 FPDE 初值问题，提出结合周期 RL/Caputo 算子、Fourier 配点与 Gegenbauer 求积的伪谱方法，将其化为良态特殊线性系统并分析存储、误差与收敛性。

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Δ)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$.

Robust BPX Preconditioner for Fractional Laplacians on Bounded Lipschitz Domains

We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power. The results apply also to the spectral and censored fractional Laplacians.

非局部扩散算子与分数拉普拉斯算子的相关比较研究

A comparative study on nonlocal diffusion operators related to the fractional Laplacian

本文研究了四种非局部扩散算子，包括分数拉普拉斯算子、谱分数拉普拉斯算子、区域分数拉普拉斯算子和周期动力学算子，通过数值实验比较它们在有界域上的差异，并发现这些算子在α趋近于2时退化为经典拉普拉斯算子，同时谱分数拉普拉斯算子的第k个特征值始终大于其他两种算子。

通过多项调和扩展近似高阶谱分数拉普拉斯幂

Approximation of higher-order powers of the spectral fractional Laplacian via polyharmonic extension

本文提出通过多项调和扩展方法近似高阶谱分数拉普拉斯幂，用于高效离散化分数阶微分算子。

Isogeometric Collocation Method for the Fractional Laplacian in the 2D Bounded Domain

We consider the isogeometric analysis for fractional PDEs involving the fractional Laplacian in two dimensions. An isogeometric collocation method is developed to discretize the fractional Laplacian and applied to the fractional Poisson problem and the time-dependent fractional porous media equation. Numerical studies exhibit monotonous convergence with a rate of $\mathcal{O}(N^{-1})$, where $N$ is the degrees of freedom. A comparison with finite element analysis shows that the method enjoys higher accuracy per degree of freedom and has a better convergence rate. We demonstrate that isogeometr…

依赖于分数阶的径向基函数无网格方法求解积分分数阶拉普拉斯算子

Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian

本文基于分数阶依赖的广义多二次径向基函数推导伪谱公式，提出一种求解积分分数阶拉普拉斯算子及分数阶扩散方程的简单高效无网格方法，并通过数值实验验证了其准确性与效率。

Two-level error estimation for the integral fractional Laplacian

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement.

一种用于谱分数拉普拉斯算子的半解析对角化有限元方法

A Semi-Analytic Diagonalization FEM for the Spectral Fractional Laplacian

本文提出一种基于Caffarelli-Silvestre扩展和对角化的半解析有限元方法，利用扩展维特征值问题的解析解来近似求解谱分数拉普拉斯算子，并验证了其性能。

分数阶拉普拉斯算子的一个新定义

A new definition of the fractional Laplacian

针对分数阶拉普拉斯算子标准定义中的超奇异积分与边界条件实施难题，提出新定义以克服这些缺陷。

Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.

分数离散拉普拉斯算子与离散化分数拉普拉斯算子

Fractional discrete Laplacian versus discretized fractional Laplacian

本文研究了离散拉普拉斯算子的分数次幂性质，并比较了其与连续分数拉普拉斯算子的离散化形式在小h下的误差估计，推导了Hölder估计及离散分数积分的性质。