A fourth-order fractional Adams-type implicit–explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions

Based on piecewise cubic interpolation polynomials, we develop a high-order numerical method with nonuniform meshes for nonlinear fractional ordinary differential equations (FODEs) with weakly singular solutions. This method is a fractional variant of the classical Adams-type implicit–explicit method widely used for integer order ordinary differential equations. We rigorously prove that the method is unconditionally convergent under the local Lipschitz condition of the nonlinear function, and when the mesh parameter is properly selected, it can achieve optimal fourth-order convergence for weakly singular solutions. We also prove the stability of the method, and discuss the applicability of the method to multi-term nonlinear FODEs and systems of multi-order nonlinear FODEs with weakly singular solutions. Numerical results are given to confirm the theoretical convergence results.