Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation

TL;DR

This paper applies Rothe's time-discretisation method to solve inverse problems for a semilinear pseudo-parabolic equation, establishing existence, uniqueness, and providing a numerical scheme with demonstrated stability and accuracy.

Contribution

It introduces a novel application of Rothe's method to inverse source problems in pseudo-parabolic equations, including a numerical scheme and stability analysis.

Findings

Proved existence and uniqueness of solutions under small data conditions.

Developed a numerical scheme based on perturbation and variational problems.

Validated the scheme's accuracy and stability through numerical experiments.

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.