In this research, by utilizing the concept of the mixed Caputo fractional derivative and left-sided mixed Riemann–Liouville fractional integral, we approximate the solution of generalized fractional Benjamin–Bona–Mahony–Burgers equations (GF-BBMBEs). In addition, using Genocchi polynomial properties, we obtain a new formula to approximate the functions by Genocchi polynomials. In the process of computation, we discuss a method of obtaining the operational matrix of integration and pseudo-operational matrices of the fractional order of derivative. Also, an algorithm of obtaining the mixed fractional integral operational matrix is presented. Using the collocation method and matrices introduced, the proposed equations are converted to a system of nonlinear algebraic equations with unknown Genocchi coefficients. In addition, we discuss the upper bound of the error for the proposed method. Finally, we examine several problems to demonstrate the validity and applicability of the proposed method.

在这项研究中，通过利用混合Caputo分数阶导数和左侧混合Riemann-Liouville分数积分的概念，我们近似了广义分数阶Benjamin-Bona-Mahony-Burgers方程（GF-BBMBEs）的解。此外，利用Genocchi多项式的性质，我们获得了一个新的公式来近似Genocchi多项式的函数。在计算过程中，我们讨论了一种获取积分运算矩阵和微分阶数伪运算矩阵的方法。此外，提出了一种获得混合分数积分运算矩阵的算法。使用配置方法和引入的矩阵，将所提出的方程转换为具有未知Genocchi系数的非线性代数方程组。此外，我们讨论了所提出方法的误差上限。最后，我们研究了几个问题，以证明所提方法的有效性和适用性。