分数阶常微分方程初值问题的高阶近似

对于整数阶常微分方程的数值解法,如欧拉法、线性多步法等都已有较完善的理论.而对于分数阶微分方程数值方法和误差估计的理论研究相对较少.在这篇文章中,我们考虑最简单的分数阶常微分方程,引进了分数阶的线性多步法,导出了分数阶常微分方程初值问题的高阶近似,证明了其方法的相容性和收敛性,并且给出了稳定性分析.最后给出了一些数值例子,证实了这个分数阶线性多步法是解分数阶常微分方程的一个有效方法.

High Order Approximations for the Fractional Ordinary Differential Equation with Initial Value Problem

Numerical method of integral order ordinary differential equation, for example, Euler method, linear multiple step method, and so on, has had quite perfect theories. Theoretical studies of the numerical method and the error estimate of fractional order differential equation are very little. In this paper, the simplest fractional order ordinary differential equation is considered. A fractional order linear multiple step method is introduced, a high order approximation of fractional order ordinary differential equation with initial value is derived, and the consistence, convergence and stability of the method are proved. Finally, some numerical examples are provided to show that the fractional order linear multiple step method for solving the fractional order ordinary differential equation is an effective method.