The main idea of this paper is to establish the novel fractional Gegenbauer functions (FGFs) for solving three kinds of fractional differential equations generated by the variable-order fractional derivatives in the Atangana-Baleanu-Caputo (ABC) sense. The numerical scheme is discussed based on the modified operational matrices (MOMs) of Atangana-Baleanu variable-order (AB-VO) fractional integration and the delay operational matrix. The methodology of obtaining the MOMs of integration is calculated with high accuracy. So that the precision of the computation method is influenced directly by the proposed matrix. In addition, we investigate the error analysis of the proposed approach. At last, several numerical experiments are employed to clarify the performance and efficiency of the method.

本文的主要思想是建立新颖的分数Gegenbauer函数（FGFs），以解决Atangana-Baleanu-Caputo（ABC）意义上的变阶分数导数生成的三种分数微分方程。基于Atangana-Baleanu可变阶（AB-VO）分数积分的修正运算矩阵（MOM）和延迟运算矩阵，讨论了数值方案。获得集成MOM的方法是高精度计算的。因此，所提出的矩阵直接影响了计算方法的精度。此外，我们调查了该方法的误差分析。最后，通过几个数值实验来阐明该方法的性能和效率。