Need help reproducing the figures from a scientific article

Hello everyone,

I am currently working on reproducing the numerical results of the following article:

Boukhari B., Mtiri F., Bchatnia A. and Beniani A. (2024), Fractional derivative boundary control in coupled Euler-Bernoulli beams: stability and discrete energy decay, AIMS Mathematics, 9(11), 32102–32123.

math-09-11-1541_251112_171636 .pdf

My objective is to reproduce, in MATLAB, Figures 2 to 7, which represent the decay of the discrete energy of the coupled Euler-Bernoulli beams system with fractional damping at the boundaries. The numerical parameters used in the article are notably:

• α = 0.5 (then α = 0.75 and α = 0.95 for certain figures);
• β = 1;
• η = 1;
• Δt = 10⁻²;
• Δx = 10⁻¹;
• L = 1;

with the initial conditions specified in Section 5.2.

math-09-11-1541_251112_171636 .pdf

The article presents:

• the finite difference scheme;
• the discretization of the tempered Caputo derivative at the boundary;
• the construction of matrices A, B and C;
• Algorithm 1 for the calculation of φ and ψ;
• Algorithm 2 for the calculation of the discrete energy.

math-09-11-1541_251112_171636 .pdf

My problem

Despite several implementations faithful to the equations and algorithms proposed in the article, I am not able to reproduce the published curves. The results obtained in MATLAB are noticeably different from Figures 2 to 7.

I wonder whether certain implementation details were not explicit in the article, for example:

• a step of numerical stabilization;
• a particular method for assembling the matrices;
• a specific treatment of boundary conditions;
• or any other details essential to reproducing the results.

I am seeking your help to:

1. Verify whether the numerical scheme presented in the article is sufficient to reproduce the figures.
2. Identify potential missing implementation details.
3. Check the construction of matrices A, B and C.
4. Verify the implementation of the boundary condition involving the fractional derivative.
5. If possible, connect with someone who has already reproduced these results or has similar MATLAB code.

Any suggestion, remark or line of thought would be welcome.

Thank you in advance for your help.