  Motivated by Luo's combinatorial Yamabe flow on closed surfaces \cite{L1} and
Guo's combinatorial Yamabe flow on surfaces with boundary \cite{Guo}, we
introduce combinatorial Calabi flow on ideally triangulated surfaces with
boundary, aiming at finding hyperbolic metrics on surfaces with totally
geodesic boundaries of given lengths. Then we prove the long time existence and
global convergence of combinatorial Calabi flow on surfaces with boundary. We
further introduce fractional combinatorial Calabi flow on surfaces with
boundary, which unifies and generalizes the combinatorial Yamabe flow and the
combinatorial Calabi flow on surfaces with boundary. The long time existence
and global convergence of fractional combinatorial Calabi flow are also proved.
These combinatorial curvature flows provide effective algorithms to construct
hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.
