*(English)*Zbl 1118.65079

The paper begins by referring to applications of fractional order equations, along with a brief summary of the main results achieved for this type of equation in the last decade.

The authors consider the nonlinear fractional-order order differential equation (NFOODE), ${}_{0}{D}_{t}^{\alpha}y\left(t\right)=f(y,t),(t>0),n-1<\alpha \le n,{y}^{\left(i\right)}\left(0\right)={y}_{0}^{\left(i\right)},i=0,1,2,\cdots ,n-1$ where $f(y,t)$ satisfies the condition $|f({y}_{1},t)-f({y}_{2},t)|\le L|{y}_{1}-{y}_{2}|$ in $t\in [0,T]$.

Existence and uniqueness theorems for the NFOODE, by *K. Diethelm* and *N. J. Ford* [J. Math. Anal. Appl. 265, No. 2, 229–248 (2002; Zbl 1014.34003)], are stated. High order fractional linear multi-step methods ($p$-HOFLMSM) are introduced. Definitions pertaining to their consistency and stability are stated. New results relating to the consistence, convergence and stability of these methods are presented and proved. The paper concludes with numerical examples which demonstrate the computational efficiency of the $p$-HOFLMSM.

##### MSC:

65L05 | Initial value problems for ODE (numerical methods) |

65L20 | Stability and convergence of numerical methods for ODE |

26A33 | Fractional derivatives and integrals (real functions) |

34K28 | Numerical approximation of solutions of functional-differential equations |

34A34 | Nonlinear ODE and systems, general |

65R20 | Integral equations (numerical methods) |

45J05 | Integro-ordinary differential equations |

45G10 | Nonsingular nonlinear integral equations |