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Title: Why is Sinc-Interpolation with Double Exponential Transform not showing exponential convergence?

Body: Hello everyone, I’m working on numerically solving the boundary value problem:

u′′(x)−u(x)=sin⁡(πx),x∈[−1,1],u(−1)=u(1)=0u''(x) - u(x) = \sin(\pi x), \quad x \in [-1,1], \quad u(-1) = u(1) = 0

I'm applying Sinc-Interpolation with the Double Exponential (DE) transformation as described in Stenger's method. I construct the second derivative matrix D(2)D^{(2)} in the tt-domain, then transform it to the xx-domain using:

Dx(2)=diag(1ϕ′(tk)2)⋅Dt(2)−diag(ϕ′′(tk)ϕ′(tk)3)⋅Dt(1)D^{(2)}_x = \text{diag}\left( \frac{1}{\phi'(t_k)^2} \right) \cdot D^{(2)}_t - \text{diag}\left( \frac{\phi''(t_k)}{\phi'(t_k)^3} \right) \cdot D^{(1)}_t

I solve the linear system

(Dx(2)−I)u=f(D^{(2)}_x - I) u = f

after applying Dirichlet boundary conditions at x=±1x = \pm1. The exact solution is known and smooth:

u(x)=−1π2+1sin⁡(πx)u(x) = -\frac{1}{\pi^2 + 1} \sin(\pi x)

However, even after increasing NN up to 50 or more, I’m not seeing exponential decay in the maximum error. The error seems to flatten out or decrease very slowly. I suspect a subtle mistake is hiding in my implementation — either in the transform, the derivative matrices, or the collocation formulation.

Any ideas on what I might be missing? Has anyone implemented Sinc collocation with DE and observed similar issues?

Thank you in advance!

If you want, I can even translate it into French or format it as a GitHub issue. Just let me know how you'd like to post it ✨