A generalized quasi-reversibility regularization method for reconstructing source terms in a time-fractional fourth-order diffusion equation driven by fractional Brownian motion

Abstract

Abstract In this paper, we study an inverse random source problem for the fourth-order fractional diffusion equation, where the source is driven by fractional Brownian motion (fBm). For the direct problem, we establish regularity results for the solutions under specific conditions. For the inverse problem, we prove the uniqueness and instability of the reconstructed source term. To address the instability, a generalized quasi-reversibility method (GQRM) is introduced, and convergence estimates for the reconstruction errors are derived under a priori and a posteriori parameter selection rules. Finally, numerical examples are presented to validate the effectiveness of the regularization approach.