In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.

在本工作中，提出了一种使用Chebyshev级数求解具有线性泛函的非线性分数阶积分微分方程（GNFIDE）的一般形式的数值技术。带有线性泛函的推荐方程会产生延迟，比例延迟和高级非线性任意阶Fredholm-Volterra积分微分方程的一般形式。扩展了光谱搭配方法，以矩阵离散化方案研究此问题，其中分数导数以Caputo形式表征。搭配方法将给定的方程和条件转换为具有未知切比雪夫系数的代数非线性方程组。此外，我们提出了导数运算矩阵的一般形式。引入的导数运算矩阵包括特殊情况的任意阶导数和普通导数运算矩阵。据作者所知，没有其他文章讨论这一点。数值算例表明，所推荐的方法是非常有效和方便的。