In this paper, we present a numerical simulation to study a fractional-order differential system of a glioblastoma multiforme and immune system. This numerical simulation is based on spectral collocation method for tackling the fractional-order differential system of a glioblastoma multiforme and immune system. We introduce new shifted fractional-order Legendre orthogonal functions outputted by Legendre polynomials. Also, we state and derive some corollaries and theorems related to the new shifted fractional order Legendre orthogonal functions. The shifted fractional-order Legendre–Gauss–Radau collocation method is developed to approximate the fractional-order differential system of a glioblastoma multiforme and immune system. The basis of the shifted fractional-order Legendre orthogonal functions is adapted for temporal discretization. The solution of such an equation is approximated as a truncated series of shifted fractional-order Legendre orthogonal functions for temporal variable, and then we evaluate the residuals of the mentioned problem at the shifted fractionalorder Legendre–Gauss–Radau quadrature points. The accuracy of the novel method is demonstrated with several test problems.

在本文中，我们提供了一个数值模拟来研究多形性胶质母细胞瘤和免疫系统的分数阶微分系统。该数值模拟基于频谱搭配方法，用于解决多形性胶质母细胞瘤和免疫系统的分数阶微分系统。我们介绍了勒让德多项式输出的新的移位分数阶勒让德正交函数。同样，我们陈述并导出与新的移位分数阶勒让德正交函数有关的一些推论和定理。移位分数阶Legendre-Gauss-Radau搭配方法被开发用于近似多形性胶质母细胞瘤和免疫系统的分数阶微分系统。移位的分数阶Legendre正交函数的基础适用于时间离散化。该方程的解近似为时间变量的分数阶Legendre正交函数的截断序列，然后我们在分数阶Legendre-Gauss-Radau正交点处评估上述问题的残差。通过几个测试问题证明了该新方法的准确性。