The theory of mixed fractional operators is still an uncovered area in fractional modelling. These multi-sided operators result by combining two fractional derivatives with different kernels, that is, the right-sided Caputo's and the left-sided Riemann–Liouville's fractional operators in a sequential manner. Although it may capture different memory phenomenons, there is no general approach to approximate the solution of the mixed type fractional differential equations (MFDEs). In this article, the authors introduce a novel numerical technique that relies on obtaining the matrix representation of the mixed operators. This was done by considering a new extension of the shifted Chebyshev polynomials of the fifth-kind (SFCPs) with a variable domain as a basis of $L^2$ space. Then we employ the spectral collocation method together with the operational matrix to transform the MFDE into the corresponding system of algebraic equations. The convergence analysis of the proposed technique was studied in terms of the generalized Taylor's formula and the Gram determinant. Our findings facilitate further development of this theory by providing such a method of approximation.

混合分数算子的理论在分数建模中仍然是一个未被发现的领域。这些多边算子是通过将两个具有不同内核的分数阶导数相结合而产生的，即右侧的 Caputo 和左侧的 Riemann-Liouville 分数算子以顺序方式组合。虽然它可能捕捉到不同的记忆现象，但没有通用的方法来逼近混合型分数阶微分方程 (MFDE) 的解。在这篇文章中，作者介绍了一种新的数值技术，它依赖于获得混合运算符的矩阵表示。这是通过考虑以可变域为基础的第五类移位切比雪夫多项式 (SFCP) 的新扩展来完成的${\mathrm{大号}}^{\mathrm{2个}}$空间。然后，我们采用谱配置方法和运算矩阵将 MFDE 转换为相应的代数方程组。根据广义泰勒公式和格拉姆行列式研究了所提出技术的收敛性分析。我们的发现通过提供这种近似方法促进了该理论的进一步发展。