In this paper, a novel adaptive procedure for step size selection for fractional differential equations is presented. The new adaptive approach is based on the implementation of a single numerical method and uses two numerical approximations, obtained at two successive steps, to advance the computation. We define a step size selection function that allows to adapt the size of the step according to the behaviour of solution. The new approach is easy to implement and leads to a low computational cost compared to classic step doubling procedure. The reported numerical results are satisfactory and show that our adaptive approach attains more accurate results than the results obtained on uniform grids, and results as good as the step doubling procedure but with very low implementation and computational effort.

在本文中，提出了一种新颖的自适应方法，用于分数阶微分方程的步长选择。新的自适应方法基于单个数值方法的实现，并使用在两个连续步骤中获得的两个数值近似值来推进计算。我们定义了一个步长选择功能，该功能允许根据解决方案的行为来调整步长的大小。与传统的步骤加倍程序相比，该新方法易于实现，并且计算成本较低。报道的数值结果令人满意，表明我们的自适应方法比在均匀网格上获得的结果更准确，结果与步长加倍程序一样好，但是实现和计算量很小。