Spatiotemporal numerical simulation of breast cancer tumors in one-dimensional nonlinear moving boundary models via temporal-spatial spectral collocation method

Abstract

In this research article, we have simulated the solutions of three types of (classical) moving boundary models in ductal carcinoma in situ by an efficient temporal-spatial spectral collocation method. In all of these three classical models, the associated fixed (spatial) boundary equations are localized by the numerical scheme. In the numerical scheme, Laguerre polynomials and Hermite polynomials are implemented to approximate the temporal and spatial variables (of unknown solutions), respectively. Then, as a generalization of the first classical model, we have considered a space-fractional moving boundary model and then transformed it, again, to the corresponding fixed boundary space-fractional equation for a straightforward discretization. Due to the impossibility of transforming of the time-fractional moving boundary model into its fixed boundary variant, we localized the time-fractional moving boundary model directly by the proposed method. The results in this category are also very satisfactory and the accuracy is again in a spectral rate. Moreover, (temporal) multi-step version of our method is applied for the considered models and the results are very accurate with respect to the single-step one, especially when the boundary of tumor is diverging in practice. In this regard, an adaptive strategy is connected to the temporal multi-step approach for a better simulation. Extensive test problems are provided to verify the accuracy of the method, with full consideration given to iterative tools for solving the final system of nonlinear algebraic equations.