Identification and stability analysis for inverse diffusion-wave fractional problems

Inverse problems for fractional diffusion-wave equations are vital in various scientific fields, but are inherently ill-posed due to the non-local and memory effects of fractional derivatives. Resolving these challenges, particularly in reconstructing multiple unknown initial values from final time data, remains a significant mathematical and computational hurdle. This work addressed this gap by establishing the well-posedness of such inverse problems involving space-time fractional operators, specifically the fractional Caputo derivative and the fractional Laplacian. We developed a stable regularization framework based on Tikhonov's method to ensure the stability and uniqueness of the solution, despite the poorly posed nature of the problem. An efficient conjugate gradient algorithm was proposed to numerically reconstruct the unknowns, with specialized techniques to handle fractional operators effectively. Numerical experiments with exact and noisy data confirmed the robustness, accuracy, and practicality of our approach, demonstrating its potential for real-world applications in modeling anomalous diffusion, heat conduction, and structural dynamics. Our results contributed both theoretical insights and computational tools for tackling complex inverse problems in fractional systems.