In this paper, a new class of computational techniques for the numerical solution of fractional functional differential equations is discussed. The proposed technique is based on Dickson polynomials with which well-known polynomials such as Fibonacci, Lucas and Chebyshev polynomials are related with some parameters. In general, the proposed combined scheme is improved by the fractional Dickson-Tau collocation technique in which a Dickson operation matrix is constructed for fractional differentiation. Then, a genetic algorithm is used to tune the unknown parameters of the proposed methods. Moreover, the error estimates and convergence of the proposed scheme are analysed. The significance of the accuracy and low computational time of the proposed scheme is verified in several numerical examples.

在本文中，讨论了一类用于分数泛函微分方程数值解的新计算技术。所提出的技术基于迪克森多项式，众所周知的多项式如斐波那契、卢卡斯和切比雪夫多项式与一些参数相关。通常，所提出的组合方案通过分数 Dickson-Tau 搭配技术得到改进，其中 Dickson 运算矩阵是为分数微分构建的。然后，使用遗传算法来调整所提出方法的未知参数。此外，分析了所提出方案的误差估计和收敛性。在几个数值例子中验证了所提出方案的准确性和低计算时间的重要性。