Solving a fractional parabolic-hyperbolic free boundary problem which models the growth of tumor with drug application using finite difference-spectral method

TL;DR

This paper develops a numerical method combining spectral and finite difference techniques to solve a complex fractional free boundary tumor growth model involving reaction-diffusion and hyperbolic equations, providing insights into tumor dynamics.

Contribution

It introduces a novel numerical approach for solving fractional parabolic-hyperbolic free boundary problems in tumor modeling, demonstrating stability and convergence.

Findings

Method is unconditionally convergent and stable.

Numerical examples validate the theoretical analysis.

Model captures tumor growth dynamics with drug and nutrient effects.

Abstract

In this paper, a free boundary problem modelling the growth of tumor is considered. The model includes two reaction-diffusion equations modelling the diffusion of nutrient and drug in the tumor and three hyperbolic equations describing the evolution of three types of cells (i.e. proliferative cells, quiescent cells and dead cells) considered in the tumor. Due to the fact that in the real situation, the subdiffusion of nutrient and drug in the tumor can be found, we have changed the reaction-diffusion equations to the fractional ones to consider other conditions and study a more general and reliable model of tumor growth. Since it is important to solve a problem to have a clear vision of the dynamic of tumor growth under the effect of the nutrient and drug, we have solved the fractional free boundary problem. We have solved the fractional parabolic equations employing a combination of spectral and finite difference methods and the hyperbolic equations are solved using characteristic equation and finite difference method. It is proved that the presented method is unconditionally convergent and stable to be sure that we have a correct vision of tumor growth dynamic. Finally, by presenting some numerical examples and showing the results, the theoretical statements are justified.