Fractional Tikhonov regularization and error estimation in inverse source problems for Biharmonic equations: a priori and a posteriori analysis under deterministic and random perturbations

This study delves into the inverse problem of identifying unknown sources within nonhomogeneous biharmonic equations confined to a bounded domain. The inherent ill-posed nature of this problem, as per Hadamard’s definition, necessitates a solution strategy. To address this challenge, we propose the Fractional Tikhonov Regularization, aimed at stabilizing the solution and providing convergence estimates. We establish both a priori and a posteriori regularization parameter selection rules to guide the regularization process. Mean Square Error estimation in \(L^p\) $L^p$ is conducted by employing the Fourier method and Sobolev embedding techniques with respect to the observed data. Additionally, convergence results are presented, specifically in scenarios where the observed data follows a Gaussian white noise distribution.