In this paper, we investigate the tumor instability by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al 2023. Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension (1D) and two dimensions (2D), we extend our analysis to three dimensions (3D). Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional framework by implementing the Alternating Directional Implicit (ADI) method, as detailed in Witelski and Bowen (2003). Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. These alignment is essential for accurately determining the stability or instability of tumor boundaries.

翻译：本文通过解析与数值方法研究肿瘤不稳定性，旨在验证先前结果并扩展Feng等人(2023)前期研究中提出的分析结论。基于对上述研究中一维(1D)及二维(2D)关键结果的重构性分析，我们将研究维度拓展至三维(3D)。具体而言，采用摄动法、渐近分析及球谐函数确定边界不稳定性。同时，通过实施Witelski与Bowen(2003)详述的交替方向隐式法(ADI)，在二维框架下验证了我们的解析结果。我们重点确保传播速度的数值模拟与解析结果精确匹配，并基于演化函数（后续章节将明确定义）得出的解析预测开展边界稳定性对比验证。这种对齐对于准确判定肿瘤边界的稳定性或不稳定性至关重要。