For terminal value problems of fractional differential equations of order $\alpha \in (0,1)$ that use Caputo derivatives, shooting methods are a well developed and investigated approach. Based on recently established analytic properties of such problems, we develop a new technique to select the required initial values that solves such shooting problems quickly and accurately. Numerical experiments indicate that this new proportional secting technique converges very quickly and accurately to the solution. Run time measurements indicate a speedup factor of between 4 and 10 when compared to the standard bisection method.

翻译：对于使用Caputo导数的阶数$\alpha \in (0,1)$的分数阶微分方程终值问题，打靶法是一种成熟且经过充分研究的方法。基于近期建立的此类问题的解析特性，我们开发了一种新技术来选择所需的初始值，能够快速准确地求解此类打靶问题。数值实验表明，这种新的比例割线技术能够快速且精确地收敛到解。运行时间测量显示，与标准二分法相比，加速因子介于4到10之间。