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

## TL;DR

This paper introduces a new finite difference and collocation method for solving a complex nonlinear free boundary problem modeling arterial plaque growth, demonstrating stability, convergence, and efficiency through numerical results.

## Contribution

It presents a novel combination of front fixing, finite difference, and collocation methods for a nonlinear free boundary problem in atherosclerosis modeling, with proven stability and convergence.

## Key findings

- Method is stable and convergent.
- Numerical results confirm efficiency.
- Effective in modeling plaque growth.

## Abstract

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.