In this article, we introduce a numerical technique for solving a class of multi-term variable-order fractional differential equation. The method depends on establishing a shifted Jacobi operational matrix (SJOM) of fractional variable-order derivatives. By using the constructed (SJOM) in combination with the collocation technique, the main problem is reduced to an algebraic system of equations that can be solved numerically. The bound of the error estimate for the suggested method is investigated. Numerical examples are introduced to illustrate the applicability, generality, and accuracy of the proposed technique. Moreover, many physical applications problems that have the multi-term variable-order fractional differential equation formulae such as the damped mechanical oscillator problem and Bagley-Torvik equation can be solved via the presented method. Furthermore, the proposed method will be considered as a generalization of many numerical techniques.

在本文中，我们介绍了一种数值技术，用于解决一类多项变量阶分数阶微分方程。该方法取决于建立分数阶可变阶导数的移位Jacobi运算矩阵（SJOM）。通过将构造的（SJOM）与并置技术结合使用，主要问题被简化为可以数值求解的代数方程组。研究了所建议方法的误差估计范围。通过数值算例说明了所提出技术的适用性，通用性和准确性。此外，通过所提出的方法可以解决许多具有多项式可变阶分数阶微分方程公式的物理应用问题，例如阻尼机械振荡器问题和Bagley-Torvik方程。此外，所提出的方法将被认为是许多数值技术的概括。