Lijuan Nong, Chen Wang, Meng Wang, Chen An
This study focuses on designing a high-order accurate method to solve the nonlinear fractional differential equation with the Caputo derivative of fractional order $\alpha\in(1, 2)$. First, we design two third-order accurate numerical approximations using the graded meshes for the fractional integral, and then employ them to establish the implicit-explicit time-stepping scheme for the equivalent Volterra integral form of the original equation. The error estimates of the two third-order numerical approximations, as well as the stability and the error estimates of the proposed time-stepping scheme, are presented rigorously. This method is further applied to solve a class of multi-term differential equations as well as a nonlinear time-fractional diffusion–wave equation. Numerical examples are extensively provided to demonstrate the effectiveness of the proposed schemes.