Implicit-Explicit difference schemes for nonlinear fractional   differential equations with non-smooth solutions
  We propose second-order implicit-explicit (IMEX) time-stepping schemes for nonlinear fractional differential equations with fractional order $0<\beta<1$. From the known structure of the non-smooth solution and by introducing corresponding correction terms, we can obtain uniformly second-order accuracy from these schemes. We prove the convergence and linear stability of the proposed schemes. Numerical examples illustrate the flexibility and efficiency of the IMEX schemes and show that they are effective for nonlinear and multi-rate fractional differential systems as well as multi-term fractional differential systems with non-smooth solutions. 
Wanrong Cao, Fanhai Zeng, Zhongqiang Zhang and George Em Karniadakis
math.NA