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端值问题，打靶法是一种已得到充分发展且被广泛研究的方法。基于此类问题近期建立的解析性质，我们提出了一种新技术来选取所需的初始值，从而快速准确地求解此类打靶问题。数值实验表明，这种新的比例割线法能够非常快速且准确地收敛到解。运行时间测量显示，与标准二分法相比，其加速比达到4至10倍。