Error analysis of finite difference/collocation method for the nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery

The main target of this paper is to present a new and efficient method to solve a nonlinear free boundary mathematical model of atherosclerosis. This model consists of three parabolics, one elliptic and one ordinary differential equations that are coupled together and describe the growth of a plaque in the artery. We start our discussion by using the front fixing method to fix the free domain and simplify the model by changing the mix boundary condition to a Neumann one by applying suitable changes of variables. Then, after employing a nonclassical finite difference and the collocation method on this model, we prove the stability and convergence of methods. Finally, some numerical results are considered to show the efficiency of the method.Comment: 35 pages, 14 Figure

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

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.Comment: 30 pages, 5 figure