In this paper, we formulate a numerical method to find out the approximate solution for fractional integro-differential equations of variable order (FIDE-VO). The methodology that adopted here is converting the FIDE-VO problem into a system of ordinary differential equations and that has been transformed into a system of algebraic equations in the unknown coefficients. For this purpose, the shifted Vieta–Fibonacci polynomials will be used for constructing both new fractional variable-order operational matrices of differentiation and integration. The variable order operators of differentiation and integration will be used in the Caputo and Riemann–Liouville senses, respectively. The Tau method and the constructed operational matrices will be used at the collocation points for transforming the FIDE-VO into an algebraic system of equations that will be solved numerically. At the end, the applicability and accuracy of the recommended method will be demonstrated through some numerical applications.

在本文中，我们制定了一种数值方法来找出分数阶积分微分方程（FIDE-VO）的近似解。此处采用的方法是将FIDE-VO问题转换为常微分方程组，并且已转换为未知系数的代数方程组。为此，移位的Vieta-Fibonacci多项式将用于构造新的微分和积分分数阶可变阶运算矩阵。微分和积分的可变阶算子将分别在Caputo和Riemann-Liouville的意义上使用。Tau方法和构造的运算矩阵将在并置点处使用，以将FIDE-VO转换为方程式的代数系统，将对其进行数值求解。最后，将通过一些数值应用论证推荐方法的适用性和准确性。