*(English)*Zbl 1060.65070

The paper presents the numerical solution of multi-order fractional differential equations of the general (possibly nonlinear) form

with $\alpha >{\beta}_{n}>{\beta}_{n-1}>\xb7\xb7\xb7>{\beta}_{1}$ and $\alpha -{\beta}_{n}\le 1$, ${\beta}_{j}-{\beta}_{j-1}\le 1$ for all $j$ and $0<{\beta}_{1}\le 1$. Its linear case is ${y}^{\left(\alpha \right)}\left(t\right)={\lambda}_{o}y\left(t\right)+{\sum}_{j=1}^{n}{\lambda}_{j}{y}^{\left({\beta}_{j}\right)}\left(t\right)+f\left(t\right)$. The initial conditions have the form ${y}^{k}\left(t\right)={y}_{o}^{\left(k\right)}$, $k=0,1,\cdots ,\lceil \alpha \rceil -1$. The derivatives are understood in the Caputo sense.

A generalization of an approach employed in the solution of ordinary differential equations of order two or higher converting such equation to a system of equations of order one is used. It uses the fact that any real number can be approximated arbitrarily closely by a rational number. Thereby, the assumption on the commensuracy for fractional order equations can be ensured by an appropriate order approximation. A simple generalization of the theorem on the equivalence of a nonlinear system and the linear systems theory is presented first. Then, the nonlinear problem includes two Gronwall-type results for a two-term equation, the general existence-uniqueness as well as the structural stability results. A convergent and stable Adams-type numerical method is proposed including a specific numerical example.

##### MSC:

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

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

34A34 | Nonlinear ODE and systems, general |

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

65L06 | Multistep, Runge-Kutta, and extrapolation methods |