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：分差方程式 $L ⁇ beta u = f$ 在紧凑度图中显示 $\ beta ⁇ frac14$ 和 $L =\ kappa =\ kappa -\ frac\ mathrm{d\ mathrm{d} x} (H\\ flac\\ matthrm{d\ d}) 是一个二级的椭圆操作器,配有某些顶端条件和足够顺畅和正系数 $\ kppappa,H$。 我们证明对于一般的顶端条件类别存在一种独特的解决方案, 并得出在Kirchhoff 顶端条件的具体情况下的解决方案的规律性2\ $\ flac\ 和 $ $\ $\ = maqalcrealal decrial=Gral decregial_ droisal dealal ral_ droisal ex.