In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.

中文翻译：

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关于具有分数时间导数的亚扩散肿瘤生长模型

在这项工作中，我们提出并分析了一个耦合偏微分方程系统，该系统模拟了在细分、机械效应、营养供应和化疗的影响下的肿瘤生长。系统的子扩散由控制肿瘤细胞体积分数的方程中的时间分数导数建模。营养物质和化疗剂的质量密度由反应扩散方程建模。我们通过 Faedo-Galerkin 方法和适当紧致性定理的应用证明了模型弱解的存在性和唯一性。最后，我们提出了一个完全离散的系统，并在数值例子中说明了分数导数的影响和分数参数的影响。