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

Highlights：

• A nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery, using the finite difference-collocation method, is solved.

• We have fixed the domain using the front fixing method and the model is simplified by changing the mixed boundary condition to a Neumann one by applying a suitable change of variables to achieve more comfortable results for numerical analysis.

• Applying the finite difference method using the second order backward difference formula (BDF2), we have constructed a sequence, which converges to the exact solution of coupled partial differential equations.

• In each time step, using Taylor theorem, the problem has changed to linear one and using the collocation method, the model is solved numerically and the convergence of the collocation method is studied.

• We have proved constructed sequence converges to the exact solution of the problem and also the stability of the method is proven.

• Some illustrative numerical examples are employed showing the efficiency of the presented method.

• We have simulated the model using finite difference and collocation methods for some pair of initial concentrations of LDL and HDL in the blood to show the validity and efficiency of the presented method.

摘要

•A nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery, using the finite difference-collocation method, is solved.•We have fixed the domain using the front fixing method and the model is simplified by changing the mixed boundary condition to a Neumann one by applying a suitable change of variables to achieve more comfortable results for numerical analysis.•Applying the finite difference method using the second order backward difference formula (BDF2), we have constructed a sequence, which converges to the exact solution of coupled partial differential equations.•In each time step, using Taylor theorem, the problem has changed to linear one and using the collocation method, the model is solved numerically and the convergence of the collocation method is studied.•We have proved constructed sequence converges to the exact solution of the problem and also the stability of the method is proven.•Some illustrative numerical examples are employed showing the efficiency of the presented method.•We have simulated the model using finite difference and collocation methods for some pair of initial concentrations of LDL and HDL in the blood to show the validity and efficiency of the presented method.