中文摘要：

经典的block-by-block方法是求解积分方程的一种高效的数值方法.研究者们已经把经典的block-by-block方法成功地用在构造非线性分数阶常微分方程的高阶数值格式上,对该格式的收敛性分析也已经有了初步的结果.但数值实验的结果表明目前的理论分析仍未达到最优阶误差估计.本文将利用Taylor公式和积分中值定理对非线性分数阶常微分方程的block-by-block方法的收敛性进行细致的分析,对其获得了最优阶误差估计,最后通过数值算例验证了理论分析的正确性.

英文摘要：

The classic block-by-block method is a highly efficient numerical method to solve the integral equation. Using the classic block-by-block method, researchers have successfully constructed higher order numerical methods for nonlinear fractional ordinary differential equ- ation, and made preliminary analysis on the convergence of this numerical method. But the results of numerical experiments show that the theoretical analysis does not achieve the optimal error estimate order. Based on the Taylor formula and integral mean value theorem, this article makes a thorough analyses on the block-by-block method of nonlinear fractional ordinary differential equations and obtains the optimal error estimate order. Finally numerical experiments are carried out to support the theoretical claims.