Jacobian-predictor-corrector approach for fractional differential equations

We present a novel predictor-corrector method, called Jacobian-predictor-corrector approach, for the numerical solutions of fractional ordinary differential equations, which are based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t. the Jacobi-weight function $\omega (s)=(1-s)^{\alpha -1} (1+s)^{0}$. This method has the computational cost O(NE) and the convergent order NI, where NE and NI are, respectively, the total computational steps and the number of used interpolation points. The detailed error analysis is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.