Identifying the Inverse Initial Condition for the Beam Equation Using Final-Time Condition

Abstract

This paper explores an inverse problem related to the initial condition of the beam equation “a fourth-order partial differential equation” with mixed boundary conditions. The goal is to recover an unknown initial condition function using additional displacement data at the final
time. We consider two types of inverse problems: one focusing on the inverse initial displacement and the other on the inverse initial velocity. This inverse initial problem is linear with a unique solution, but it is considered ill-posed because solutions may not exist for all input data. Even when solutions do exist, they might not depend continuously on the input. To achieve a stable numerical solution, we apply the finite difference method combined with Tikhonov regularization, testing various regularization orders. The regularization parameter is determined using the L-curve method and the norm error. Our numerical results demonstrate that this approach yields accurate solutions for exact data and stable results for noisy data. Although the literature describes conditions ensuring uniqueness and continuous dependence, it lacks numerical evidence. Thus, developing an efficient numerical method for this linear yet ill-posed problem is our main goal and contribution.