求解多项分数阶常微分方程的数值方法

Numerical Methods for Multi-terms Fractional-Order Ordinary Differential Equation

本文考虑多项的分数阶常微分方程。证明了其解的存在性与唯一性;导出了多项的分数阶常微分方程的解;提出了三种数值解法来近似多项的分数阶常微分方程解。第一种方法,利用Diethelm等技巧;第二种方法,利用Caputo分数阶导数,Riemann-Liouville分数阶导数,分数阶导数之间的关系;第三种方法,把多项的分数阶常微分方程转化为分数阶微分方程组,然后利用分数阶预估-校正法。最后给出了一些实际应用例子。

In this paper, a multi -- terms fractional - order ordinary differential equation is considered. The existence and uniqueness of the solution of this model is proved, and the analytical solution of the multi - terms fractional - order ordinary differential equation is derived. Three ntnnerical methods for solving the multi - terms fractional - order ordinary differential equation are proposed. The first method use Diethelm et al. ＇s technique; The second method use the relationship between the Captro, Riemann - IAouville and fractional - order derivatives; In the third method, the multi - terms fi-aetional - order ordi- nary differential equalicn is tranferred into a system of fractional order differential equations. A new computationally effective fractional Predictor- Corrector maethod is proposed for simulating the fi-actional order systems. Finally, some exam- pies of practical applicatiom are presented.