In this work, a study of the mathematical model of uncontrolled growth of tumor cells in the presence of chemotherapy is proposed. The fractional form of the model with the non-singular exponential kernel is considered. This model consists of four coupled partial differential equations (PDEs) which depict the relationship among the normal, tumor, immune cells, and the chemotherapy parameter. The purpose is to investigate the behavior of all types of cells with a change in the fractional order parameter and to show the effect of chemotherapeutic treatment on tumor cells with different levels of immune system. Before applying the proposed numerical method, an approximate expression for the fractional-order Caputo-Fabrizio (C-F) derivative of polynomial function $x^k$ is derived. A shifted Chebyshev operational matrix of fractional order derivative is deduced by using an approximation of the C-F fractional derivative of the function and some properties of the orthogonal polynomials. The system of four coupled fractional order PDEs is studied by using the operational matrix method. The dynamics of all the aforementioned cells with respect to different fractional order derivatives are derived and computed numerically for the prescribed values of parameters. These are depicted through graphs to study the diffusive nature of cells and the effect of chemotherapy on all types of cells, before and after applying the therapy. This study shows that tumor cell growth decreases with time when chemotherapy treatment is started. The concentration of tumor cells is more in the invasive fronts of the tumor site as compared to the center of the tumor. It is concluded that the growth of tumor cells is less due to chemotherapy treatment for a person with a strong immune system.

在这项工作中，提出了对在化学疗法存在下肿瘤细胞不受控制的生长的数学模型的研究。考虑具有非奇异指数核的模型的分数形式。该模型由四个耦合的偏微分方程（PDE）组成，这些方程描述了正常，肿瘤，免疫细胞和化疗参数之间的关系。目的是研究分数阶参数变化的所有类型细胞的行为，并显示化学治疗对免疫系统水平不同的肿瘤细胞的影响。在应用所提出的数值方法之前，需要多项式函数的分数阶Caputo-Fabrizio（CF）导数的近似表达式$X^{ķ}$派生。通过使用函数的CF分数导数和正交多项式的某些性质的逼近来推导分数阶导数的偏移Chebyshev运算矩阵。利用运算矩阵方法研究了四个耦合的分数阶PDE系统。对于参数的规定值，针对不同的分数阶导数，得出所有上述单元的动力学并进行数值计算。这些通过图表进行了描述，以研究细胞在应用治疗之前和之后的扩散特性以及化学疗法对所有类型细胞的影响。这项研究表明，开始化疗后，肿瘤细胞的生长会随着时间的推移而减少。与肿瘤中心相比，肿瘤细胞的浓度在肿瘤部位的侵袭性前沿更高。结论是，对于具有强免疫系统的人，由于化学疗法的治疗，肿瘤细胞的生长较少。