Numerical Solution of the Coupled System of Nonlinear Fractional Ordinary Differential Equations (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we consider the numerical method that is proposed and analyzed in [J. Cao and C. Xu, J. Comput. Phys., 238 (2013), pp. 154–168] for the fractionalordinary differential equations. It is based on the so-called block-by-block approach,which is a common method for the integral equations. We extend the technique tosolve the nonlinear system of fractional ordinary differential equations (FODEs) andpresent a general technique to construct high order schemes for the numerical solutionof the nonlinear coupled system of fractional ordinary differential equations (FODEs).By using the present method, we are able to construct a high order schema for nonlinear system of FODEs of the order $α$, $α>0$. The stability and convergence of the schemais rigorously established. Under the smoothness assumption $f,g ∈ C^4[0,T]$, we provethat the numerical solution converges to the exact solution with order $3+α$ for $01$. Some numerical examples are provided to confirm the theoreticalclaims.