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We propose second-order implicit-explicit (IMEX) time-stepping schemes fornonlinear fractional differential equations with fractional order $0<\beta<1$.From the known structure of the non-smooth solution and by introducingcorresponding correction terms, we can obtain uniformly second-order accuracyfrom these schemes. We prove the convergence and linear stability of theproposed schemes. Numerical examples illustrate the flexibility and efficiencyof the IMEX schemes and show that they are effective for nonlinear andmulti-rate fractional differential systems as well as multi-term fractionaldifferential systems with non-smooth solutions.

Implicit-Explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions