The fractional differential equation $L^\beta u = f$ posed on a compact metric graph is considered, where $\beta>0$ and $L = \kappa^2 - \nabla(a\nabla)$ is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients $\kappa, a$. 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 ${L = \kappa^2 - \Delta, \kappa>0}$ are performed to illustrate the results.

翻译：考虑定义在紧致度量图上的分数阶微分方程 $L^\beta u = f$，其中 $\beta>0$，$L = \kappa^2 - \nabla(a\nabla)$ 为二阶椭圆算子，配备特定顶点条件且系数 $\kappa, a$ 充分光滑且正值。本文证明了在一般顶点条件下解的存在唯一性，并在基尔霍夫顶点条件的特例下推导了解的正则性。这些结果被推广至 $f$ 替换为高斯白噪声的随机设定。对于广义基尔霍夫顶点条件下的确定性与随机设定，我们提出基于有限元逼近与分数阶幂 $L^{-\beta}$ 有理逼近的数值解法。针对所得逼近，分析了确定性情形下的强误差，以及在随机情形下解的协方差函数的强均方误差与 $L_2(\Gamma\times \Gamma)$-误差。所有情形均推导出显式收敛速率。通过 ${L = \kappa^2 - \Delta, \kappa>0}$ 的数值实验验证了理论结果。