Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation
Abstract
In this paper, we investigate the inverse problem of determining an unknown time-dependent source term in a semilinear pseudo-parabolic equation with variable coefficients and a Dirichlet boundary condition. The unknown source term is recovered from additional measurement data expressed as a weighted spatial average of the solution. By employing Rothe's time-discretisation method, we prove the existence and uniqueness of a weak solution under a smallness condition on the problem data. We also provide a numerical scheme based on a perturbation approach, which reduces the solution of the resulting discrete problem to solving two standard variational problems and evaluating a scalar coefficient, and we demonstrate its accuracy and stability through numerical experiments.
Keywords
Cite
@article{arxiv.2510.20642,
title = {Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation},
author = {Karel Van Bockstal and Khonatbek Khompysh and Arshyn Altybay},
journal= {arXiv preprint arXiv:2510.20642},
year = {2026}
}