  In this paper, an efficient algorithm is presented by the extrapolation
technique to improve the accuracy of finite difference schemes for solving the
fractional boundary value problems with non-smooth solution. Two popular finite
difference schemes, the weighted shifted Gr\"{u}nwald difference (WSGD) scheme
and the fractional centered difference (FCD) scheme, are revisited and the
error estimate of the schemes is provided in maximum norm. Based on the
analysis of leading singularity of exact solution for the underlying problem,
it is demonstrated that, with the use of the proposed algorithm, the improved
WSGD and FCD schemes can recover the second-order accuracy for non-smooth
solution. Several numerical examples are given to validate our theoretical
prediction. It is shown that both accuracy and convergence rate of numerical
solutions can be significantly improved by using the proposed algorithm.
