Fractional high order methods for the nonlinear fractional ordinary differential equation

In this paper, we consider the nonlinear fractional-order ordinary differential equation 0Dtαy(t)=f(y,t), (t>0)(t>0), n−1<α≤nn−1<α≤n, y(i)(0)=y0(i), i=0,1,2,…,n−1i=0,1,2,…,n−1, where f(y,t)f(y,t) satisfies the LL-condition, i.e., |f(y1,t)−f(y2,t)|≤L|y1−y2||f(y1,t)−f(y2,t)|≤L|y1−y2| in t∈[0,T]t∈[0,T]. Fractional-order linear multiple step methods are introduced. The high order (2–6) approximations of the fractional-order ordinary differential equation with an initial value are proposed. The consistence, convergence and stability of the fractional high order methods are proved. Finally, some numerical examples are provided to show that the fractional high order methods for solving the fractional-order nonlinear ordinary differential equation are computationally efficient solution methods.