In this article, a novel shifted Gegenbauer operational matrix (SGOM) of fractional derivative in the Caputo sense is derived. Based on this operational matrix, an accurate and effective numerical algorithm is proposed. The SGOM of fractional derivative in conjunction with the tau method are used for solving the constant and variable coefficients space–time fractional telegraph equations (FTE) with various types of boundary conditions, namely, Neumann, Dirichlet and Robin conditions. The convergence analysis of the proposed method is established in \(\mathcal {L}^{2}_{\omega _{\alpha }}\). Finally, miscellaneous test examples are given and compared with other methods to clarify the accuracy and efficiency of the presented algorithm.

在本文中，得出了一种在Caputo意义上的分数阶导数的新型移位Gegenbauer操作矩阵（SGOM）。基于该运算矩阵，提出了一种准确有效的数值算法。分数导数的SGOM与tau方法一起用于求解具有各种边界条件（即Neumann，Dirichlet和Robin条件）的常数和可变系数时空分数电报方程（FTE）。所提出方法的收敛性分析是在\（\ mathcal {L} ^ {2} _ {\ omega _ {\ alpha}} \）中建立的。最后，给出了其他测试示例并将其与其他方法进行比较，以阐明所提出算法的准确性和效率。