Cancer belongs to the class of diseases which is symbolized by out of control cells growth. These cells affect DNAs and damage them. There exist many treatments available in medical science as radiation therapy, targeted therapy, surgery, palliative care and chemotherapy. Chemotherapy is one of the most popular treatments which depends on the type, location and grade of cancer. In this paper, we are working on modeling and prediction of the effect of chemotherapy on cancer cells using a fractional differential equation by using the differential operator in Caputo’s sense. The presented model depicts the interaction between tumor, normal and immune cells in a tumor by using a system of four coupled fractional partial differential equations (PDEs). For this system, initial conditions of tumor cells and dimensions are taken in such a way that tumor is spread out enough in size and can be detected easily with the clinical machines. An operational matrix method with Genocchi polynomials is applied to study this system of fractional PDEs (FPDEs). An operational matrix for fractional differentiation is derived. Applying the collocation method and using this matrix, the nonlinear system is reduced to a system of algebraic equations, which can be solved using Newton iteration method. The salient features of this paper are the pictorial presentations of the numerical solution of the concerned equation for different particular cases to show the effect of fractional exponent on diffusive nature of immune cells, tumor cells, normal cells and chemotherapeutic drug and depict the interaction among immune cells, normal cells and tumor cells in a tumor site.

中文翻译：

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化疗存在下肿瘤细胞非线性分数数学模型的数值研究

癌症属于以细胞生长失控为特征的一类疾病。这些细胞会影响 DNA 并破坏它们。医学科学中有许多可用的治疗方法，如放射治疗、靶向治疗、手术、姑息治疗和化疗。化疗是最流行的治疗方法之一，这取决于癌症的类型、位置和等级。在本文中，我们正在使用 Caputo 意义上的微分算子，使用分数微分方程对化疗对癌细胞的影响进行建模和预测。所提出的模型通过使用四个耦合分数偏微分方程 (PDE) 的系统来描述肿瘤中肿瘤、正常细胞和免疫细胞之间的相互作用。对于这个系统，肿瘤细胞的初始条件和尺寸以这样一种方式获取，即肿瘤的大小足够分散，并且可以用临床机器轻松检测到。使用 Genocchi 多项式的运算矩阵方法来研究这种分数 PDE (FPDE) 系统。导出了分数微分的运算矩阵。应用搭配法并使用该矩阵，将非线性系统简化为代数方程组，可以使用牛顿迭代法求解。本文的显着特点是对不同具体情况下有关方程的数值解的图形表示，以展示分数指数对免疫细胞、肿瘤细胞、正常细胞和化疗药物的扩散性质的影响，并描绘免疫细胞之间的相互作用。细胞，