多项分数阶常微分方程的数值积分法摘要 近年来，分数阶微积分在解决各种具有遗传和记忆性质的物理、材料和力学、信息领域问题中大放异彩，在建立各种数学模型的场合最为普遍使用。本文主要是通过Riemann-Liouville 分数阶积分来研究并解决多项分数阶常微分方程初值问题，比较基于一阶矩形公式的显式方法和基于二阶卷积权方法对于具体算例解的误差分析，对求解多项分数阶常微分方程的数值积分方法进行探讨。在解初值问题之前，本文首先引入了Riemann-Liouville 分数阶积分的定义及相关性质以便读者理解，给出了基于一阶左右矩形公式的显式方法和基于二阶卷积权逼近法两种求解多项分数阶常微分方程初值问题的常规方法；还演示了一例基础的三项分数阶微分方程初值问题如何用以上两种方法求解，并根据不同的步长进行多次迭代比较，对比迭代过程中的收敛阶数和近似解与精确解进行误差分析，评价了两种方法的适用环境和相关优劣性。希望找到新的理论方法，以打破现有的限制条件，力求构建一套完善的分数阶微分方程理论。关键词 多项分数阶常微分方程 基于一阶矩形公式的显式方法 二阶卷积权方法 Numerical integration method for multiple fractional ordinary differential equationsAbstract In recent years, fractional calculus has been used to solve various problems with genetic and memory characteristics in physics, materials, mechanics, information and other fields, as well as to establish various mathematical models. In this paper, Riemann-Liouville fractional integration is used to study and solve the initial value problem of multiple fractional ordinary differential equations. The error analysis of the specific solution based on the explicit method based on the first-order rectangular formula and the method based on the second-order convolution weight is compared. The numerical integration method for solving multi-terms fractional-order ordinary differential equation is discussed. Before solving the initial value problem, this paper first introduces the definition and related properties of Riemann Liouville fractional inte...
