In this paper, we propose an adaptive procedure, recently developed for fractional ordinary differential equations, for the solutions of time-fractional advection–diffusion–reaction equations involving the Caputo derivative. We focus on the adaptivity of the discretization in time direction defining a step size selection function that allows adapting the time step size according to the local behaviour of the solution. The new approach is easy to implement and reveals to have a low computational cost. Test problems are reported and comparisons with results found in literature confirm the accuracy and efficiency of the step-size selection procedure for the solutions of fractional partial differential equations.

在本文中，我们提出了一种自适应程序，最近为分数常微分方程开发，用于求解涉及 Caputo 导数的时间分数平流-扩散-反应方程。我们专注于时间方向离散化的适应性，定义了一个步长选择函数，允许根据解的局部行为调整时间步长。新方法易于实现并且显示出具有低计算成本。报告了测试问题，并与文献中的结果进行了比较，确认了分数偏微分方程解的步长选择程序的准确性和效率。