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arxiv: 1907.04097 · v1 · pith:L2JHWTQPnew · submitted 2019-07-09 · 🧮 math.NA · cs.NA

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

Farzaneh Nasresfahani , Mohammad Reza Eslahchi This is my paper

Pith reviewed 2026-05-25 00:18 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords free boundary problemfinite difference methodcollocation methodplaque growthatherosclerosisstability analysisconvergence analysisnumerical PDE
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The pith

Front-fixing transformation followed by nonclassical finite difference and collocation methods produces stable convergent solutions for the nonlinear coupled plaque growth model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a numerical method for a mathematical model of atherosclerosis consisting of three coupled parabolic equations, one elliptic equation and one ordinary differential equation that describe plaque growth with a moving free boundary. A front-fixing transformation is applied first to convert the problem to a fixed spatial domain and to replace the mixed boundary condition with a Neumann condition. The transformed system is then discretized by a nonclassical finite difference scheme together with a collocation method. The authors prove that these discretizations are stable and convergent, and they present numerical experiments that illustrate the method's performance on the full nonlinear system.

Core claim

After the front-fixing transformation and change of variables, the nonclassical finite difference and collocation methods applied to the resulting system of equations are stable and converge to the solution of the original free-boundary model.

What carries the argument

Front-fixing transformation that immobilizes the free boundary and converts the mixed boundary condition into a Neumann condition, followed by nonclassical finite difference and collocation discretization of the fixed-domain system.

If this is right

  • The discrete solutions can be used to compute the time evolution of plaque thickness and the concentrations of the biological species inside the artery wall.
  • Error bounds between the numerical approximation and the true solution follow directly from the convergence proof.
  • The same discretization framework applies to the coupled system without further modification once the transformation has been performed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same front-fixing plus collocation strategy could be tested on other free-boundary models that couple reaction-diffusion equations to an ordinary differential equation for interface motion.
  • If the convergence rates remain high under realistic parameter ranges, the scheme could support inverse problems that estimate growth coefficients from medical imaging sequences.

Load-bearing premise

The front-fixing transformation and the subsequent change of variables that converts the mixed boundary condition into a Neumann condition remain valid for the full nonlinear coupled system.

What would settle it

A sequence of numerical solutions on successively refined meshes that fails to approach a known exact solution, or that exhibits instability for parameter values inside the model's physical range, would show that the stability or convergence claim does not hold.

Figures

Figures reproduced from arXiv: 1907.04097 by Farzaneh Nasresfahani, Mohammad Reza Eslahchi.

Figure 1
Figure 1. Figure 1: Seven steps to show the role of mathematical modeling in Solving the mathematical Problems [33]. • We have proved constructed sequence converges to the exact solution of the problem (see Theorem 5.2) and also the stability of the method is proven (see Theorem 5.3). • We have simulated the model using finite difference and collocation method for some pair of values (L0, H0) to show the validity and efficien… view at source ↗
Figure 2
Figure 2. Figure 2: The process of plaque development [1]. cholesterol, macrophages and foam cells ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Maximum time-error functions with N=10 and various M by considering TFBM. Lies in the fact that it is hard or sometimes impossible to reach the exact solution of most of the cou￾pled nonlinear models analytically, and because we have shown that the presented method for solving the model is stable and convergent (See Theorem 5.2), so, we consider numerical results for the large M = 300 and N = 10 as an exac… view at source ↗
Figure 4
Figure 4. Figure 4: The behaviour of Maximum time-error with N=10 and various M in Log-Log scale by considering TFBM. Rate of convergence for M L H F 100 - - - 200 1.321 1.319 0.648 300 1.426 1.425 0.706 400 1.607 1.607 1.159 500 1.874 1.873 1.546 600 2.271 2.270 2.012 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Maximum space-error functions with M=100 and various N by considering TFBM. highlighted in bold. Because of the fact that the condition number of the coefficient matrices grows very fast when N > 10, unfortunately, the Matlab software can not accurately extract the results. To overcome this difficulty, we need to propose proper trial functions which reduce the condition number significantly. In this case, … view at source ↗
Figure 6
Figure 6. Figure 6: Maximum time-error with N=50 and various M by considering TFBL. Thus (72) becomes as follows pn(x) = Ln(x) − n(n + 1) (n + 2)(n + 3)Ln+2(x), n ≥ 0. (75) So we can approximate the functions L(ρ, t), H(ρ, t), F(ρ, t) in the form of (37) as follows L N n+1(ρ) = X N i=0 l n+1 i pi(ρ), HN n+1(ρ) = X N i=0 h n+1 i pi(ρ), F N n+1(ρ) = X N i=0 f n+1 i pi(ρ), where pi(ρ) = Li(ρ) − (i + 1)i (i + 3)(i + 2)Li+2(ρ), i … view at source ↗
Figure 7
Figure 7. Figure 7: The behaviour of maximum time error with N=50 and various M in Log-Log scale by considering TFBL. Rate of convergence for M L H F 100 - - - 200 1.118 1.179 1.206 300 1.339 1.334 1.350 400 1.542 1.539 1.550 500 1.822 1.822 1.828 600 2.228 2.226 2.223 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Maximum space error function for M=200 and various N. the problem with N = 200 and M = 200 as an exact solution. To better observe the space-error of nu￾merical results, [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of condition numbers of the coefficient matrices in both TFBM and TFBL cases. noteworthy that arrows in these two figures indicate the direction of growth or shrink of the plaque. 101 102 101 102 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Condition number of the coeffi￾cient matrices (CN). N Eq. (1) Eq. (2) Eq. (3) 10 3.245 3.245 4.096 20 17.475 17.475 7.230 40 1.302e+02 1.302e+02 13.61 80 1.022e+02 1.022e+02 26.14 100 1.990e+02 1.990e+02 32.05 [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Variation of the radius of the plaque toward the various level of L0 and H0 in blood during the days [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Risk Map. The values of LDL and HDL are measured in mg/dl = 10−4 g/cm3 [19] [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Variation of the radius of the plaque with L0 = 150 and H0 = 45 in a small part of the artery [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Variation of the radius of the plaque with L0 = 120 and H0 = 60 in a small part of the artery [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a numerical method for a nonlinear free boundary problem modeling plaque growth, consisting of three coupled parabolic PDEs, one elliptic PDE, and one ODE. It applies a front-fixing transformation to fix the domain and a change of variables to convert mixed boundary conditions to Neumann type, then employs nonclassical finite difference and collocation methods, proves stability and convergence of the methods, and presents numerical results to demonstrate efficiency.

Significance. If the transformation is shown to preserve the structure of the full nonlinear coupled system and the stability/convergence proofs are rigorous, the work provides a useful contribution to numerical analysis of free-boundary problems in mathematical biology, with potential relevance to biomedical modeling of atherosclerosis.

major comments (1)
  1. [Transformation and change of variables (preceding the method application)] The front-fixing transformation and change of variables (described prior to the numerical discretization) are presented at a high level. It is not shown explicitly that the elliptic equation and the ODE remain consistent with the original nonlinear couplings, or that no additional singularities or coupling terms are introduced that would affect the subsequent finite-difference/collocation analysis. This verification is load-bearing for the stability and convergence claims.
minor comments (1)
  1. [Proof sections] The abstract and introduction assert proofs of stability and convergence, but the manuscript would benefit from clearer cross-references between the transformed system equations and the specific estimates used in the proofs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the transformation step. We address the concern below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Transformation and change of variables (preceding the method application)] The front-fixing transformation and change of variables (described prior to the numerical discretization) are presented at a high level. It is not shown explicitly that the elliptic equation and the ODE remain consistent with the original nonlinear couplings, or that no additional singularities or coupling terms are introduced that would affect the subsequent finite-difference/collocation analysis. This verification is load-bearing for the stability and convergence claims.

    Authors: We agree that the presentation of the transformed system can be strengthened by greater explicitness. In the revised manuscript we will add a dedicated subsection that derives the fully transformed equations for all five components of the model (the three parabolic PDEs, the elliptic PDE, and the ODE). This derivation will verify term-by-term that the original nonlinear couplings are preserved, that the change of variables maps the mixed boundary conditions to Neumann conditions without introducing singularities, and that the resulting system remains suitable for the subsequent stability and convergence analysis. The proofs will then reference these explicit transformed equations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on transformed system

full rationale

The paper describes applying the front-fixing transformation and variable change to convert the free-boundary model into a fixed-domain system with Neumann conditions, followed by nonclassical finite differences and collocation, then proving stability and convergence on that transformed system. No quoted equations, fitted parameters, or self-citations reduce the stability/convergence claims to tautologies or inputs by construction. The transformation is presented as a modeling step whose validity is presupposed for the subsequent analysis; this is an assumption, not a circular reduction within the derivation chain itself. The numerical analysis is therefore independent of the original free-boundary formulation once the transformed equations are accepted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms or invented entities; the front-fixing transformation is treated as a standard technique rather than a new postulate.

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