A time-dependent inverse source problem for a semilinear pseudo-parabolic equation with Neumann boundary condition

TL;DR

This paper addresses the inverse problem of identifying a time-dependent source in a semilinear pseudo-parabolic PDE with Neumann boundary conditions, establishing theoretical results and numerical methods for solution recovery.

Contribution

It introduces a Rothe's method-based approach for proving existence and uniqueness, and develops a numerical scheme with convergence analysis for the inverse source problem.

Findings

Existence and uniqueness of weak solutions are proven.

A convergent numerical time-discrete scheme is designed.

Numerical experiments validate the theoretical results and handle noisy data.

Abstract

In this paper, we study the inverse problem for determining an unknown time-dependent source coefficient in a semilinear pseudo-parabolic equation with variable coefficients and Neumann boundary condition. This unknown source term is recovered from the integral measurement over the domain $\Omega$. Based on Rothe's method, the existence and uniqueness of a weak solution, under suitable assumptions on the data, is established. A numerical time-discrete scheme for the unique weak solution and the unknown source coefficient is designed, and the convergence of the approximations is proven. Numerical experiments are presented to support the theoretical results. Noisy data is handled through polynomial regularisation.