In this talk, we present two high-order discrete schemes and their applications for Caputo fractional derivative. In the first part, we introduce the shifted fractional trapezoidal rule (SFTR) and its key properties. Based on SFTR, we construct an efficient numerical scheme for the time-fractional Allen-Cahn equation, rigorously proving its discrete energy decay and maximum-principle preservation properties. Numerical investigations show that the scheme can resolve the intrinsic initial singularity of such nonlinear fractional equations as tFAC equation on uniform meshes without any correction. We consider the H2N2 scheme in the second part, which provides a fast finite difference algorithm for the fractional sine-Gordon equation, with proven stability and high-order convergence. Numerical experiments validate the superiority of both schemes and the correctness of the theoretical results.