A high order schema for the numerical solution of the fractional ordinary differential equations

In this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations (FODEs). This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In our approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps, while preserving the good stability property of the block-by-block schemes. By using this new approach, we are able to construct a high order schema for FODEs of the order α , α > 0 . The stability and convergence of the schema is rigorously established. We prove that the numerical solution converges to the exact solution with order 3 + α for 0 < α ⩽ 1 , and order 4 for α > 1. A series of numerical examples are provided to support the theoretical claims.