The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

第五类移位切比雪夫多项式（SCPFK）和搭配方法被用来获得一类函数方程的近似解，即变阶时间分数弱奇异偏积分微分方程（VTFWSPIDE）。一种伪运算矩阵 (POM) 方法被开发用于所研究问题的数值解。建议的方法将求解 VTFWSPIDE 更改为求解线性代数方程组。获得了近似解的误差界限，并检验了所提出方案在五个问题上的应用。结果证实了该方法对分数阶奇异偏积分微分方程数值求解的适用性和高精度。