Dear Editor,

Please consider our manuscript entitled

"Boundary-Consistent L1-Legendre Spectral Approximation of Nonlinear Caputo Reaction-Diffusion Problems with Integral Constraints"

for publication in Boundary Value Problems.

The manuscript develops and verifies a boundary-consistent discretization for nonlinear Caputo reaction-diffusion equations with non-local integral endpoint constraints. The principal issue addressed is that a boundary-adapted polynomial basis need not vanish at the endpoints. Consequently, the conventional derivative stiffness form can differ from the Galerkin representation of the strong diffusion operator by a nonzero endpoint term. We therefore assemble the strong-form matrix directly and couple it with the classical L1 approximation in time.

The manuscript provides stability and convergence results under explicit discrete accretivity, regularity, and constrained-approximation assumptions; a contractive regime for the nonlinear residual-damped iteration; multi-order temporal convergence tests; a smooth non-polynomial spectral benchmark; an accretivity scan; and a direct operator-consistency experiment. In that experiment the strong-form discretization converges spectrally to machine precision, whereas the conventional derivative-stiffness replacement stagnates at a nonzero L2 error plateau despite satisfying the same integral constraints.

The work is explicitly presented as a follow-up numerical study to the authors' earlier boundary-adapted Legendre framework. The earlier lifting and basis construction are cited and are not claimed as new contributions here. Reproducible Python code and the numerical data used for the reported verification tests are supplied.

The manuscript is original, is not under consideration elsewhere, and all authors have approved its submission.

Sincerely,

Alvaro H. Salas S. (corresponding author)
FIZMAKO Research Group
Universidad Nacional de Colombia, Sede Manizales
ahsalass@unal.edu.co
