THIRD-ORDER ACCURATE IMPLICIT-EXPLICIT TIME-STEPPING SCHEME FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER <tex-math id="M1">α∈(1, 2)</tex-math>

This study focuses on designing a high-order accurate method to solve the nonlinear fractional differential equation with the Caputo derivative of fractional order α∈(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.