DeepAI AI Chat
Log In Sign Up

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

by   Sakine Esmaili, et al.

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.


page 1

page 2

page 3

page 4


An ADI Scheme for Two-sided Fractional Reaction-Diffusion Equations and Applications to an Epidemic Model

Reaction-diffusion equations are often used in epidemiological models. I...

An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains

In this work, we apply a fast and accurate numerical method for solving ...

On Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a problem with fraction...

Finite element modelling of in-stent restenosis

From the perspective of coronary heart disease, the development of stent...

Complex far-field geometries determine the stability of solid tumor growth with chemotaxis

In this paper, we develop a sharp interface tumor growth model to study ...

Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis

In this paper, we develop a sharp interface tumor growth model to study ...