The fractional differential equation $L^\beta u = f$ posed on a compact metric graph is considered, where $\beta>\frac14$ and $L = \kappa - \frac{\mathrm{d}}{\mathrm{d} x}(H\frac{\mathrm{d}}{\mathrm{d} x})$ is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients $\kappa,H$. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when $f$ is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power $L^{-\beta}$. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the $L_2(\Gamma\times \Gamma)$-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for the example ${L = \kappa^2 - \Delta, \kappa>0}$ are performed to illustrate the theoretical results.

翻译：本文考虑定义在紧度量图上的分数阶微分方程 $L^\beta u = f$，其中 $\beta>\frac14$，$L = \kappa - \frac{\mathrm{d}}{\mathrm{d} x}(H\frac{\mathrm{d}}{\mathrm{d} x})$ 为具有特定顶点条件且系数 $\kappa,H$ 充分光滑且为正的二阶椭圆算子。我们证明了在一般顶点条件下解的唯一存在性，并在 Kirchhoff 顶点条件下推导了解的正则性。当 $f$ 替换为高斯白噪声时，这些结果被推广到随机情形。对于广义 Kirchhoff 顶点条件下的确定性与随机情形，我们提出了一种基于有限元逼近与分数幂 $L^{-\beta}$ 有理逼近相结合的数值求解方法。针对所得逼近，在确定情形下分析了强误差，在随机情形下分析了均方强误差以及解协方差函数的 $L_2(\Gamma\times \Gamma)$ 误差。所有情形均推导出显式收敛速率。以 ${L = \kappa^2 - \Delta, \kappa>0}$ 为例进行数值实验，验证了理论结果。