Jacobian-Predictor-Corrector Approach For Fractional Differential Equations

Fractional derivatives are generalizations of the classical integer-order counter-parts. Fractional operators can be used more accurately to describe the probabil-ity distribution of particles in space and in time. Recently, the fractional calculus have been widely applied in the physical and biological engineering. At the same time, they have become the most effective tools to describe the abnormal diffusion problems. The modeling progress on using fractional differential equations has led to increasing interest on numerical schemes for their solutions. 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 ω(s)=(1-s)α-1(1+s)0. During both the predict and correct procedure, the values used in the quadrature are approximated by those of local interpolate functions’. Unlike in predictor, to obtain the local interpolations in corrector, the predicted value is used. 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.