A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation.

考虑了具有Caputo分数导数的非线性分数微分方程。该问题通过逆风有限差分方案进行离散化，针对该方案导出了最大范数中的后验误差分析。基于此后验误差估计的部分启发式论证导致了几个合适的监控函数，并构造了一个新的监控函数来设计自适应网格算法。给出了数值结果来说明所提出的自适应方法的性能。与其他监控函数相比，提出的基于新监控函数的自适应网格方法更适合求解此类非线性分数阶微分方程。