Identifying the Inverse Force Problem for the Beam Equation With Boundary Conditions at a Single Point in Space or Time

We analyze an inverse force problem that involves identifying a source term in a one-dimensional fourth-order beam equation depending on either space or time. Mixed boundary conditions and an overdetermination condition based on a single point in space or time address the problem. For both direct and inverse problems, we rely on the existence and uniqueness theorems from previous research. Likewise, we apply the Von Neumann method to determine stability. The finite difference method (FDM) with Tikhonov regularization effectively identifies displacement and source problems in numerical analysis. Consequently, a small amount of noise improves the accuracy and consistency of the final results, as demonstrated by tests conducted at various noise levels. Numerical experiments with both exact and noisy data show the accuracy, reliability, and efficiency of our approach.