Multi-operator iterative regularization framework for caputo-Hadamard fractional diffusion with environmental applications

We present a novel Fractional Power Iterative regularization method for solving Caputo-Hadamard fractional inverse source problems arising in anomalous diffusion applications. The proposed method addresses the inherent ill-posedness of reconstructing unknown source terms from noisy boundary observations by combining fractional power regularization operators with iterative refinement strategies. Rigorous convergence analysis establishes superlinear convergence rates and optimal error bounds under appropriate smoothness assumptions, with spectral analysis revealing exponential decay characteristics that significantly outperform classical Tikhonov regularization. Comprehensive numerical experiments demonstrate the method’s superiority across multiple performance metrics, showing substantial improvements in reconstruction accuracy and optimal linear scaling behavior with noise levels, while maintaining exceptional spectral preservation capabilities even under challenging noise conditions. Environmental applications to groundwater contamination transport modeling demonstrate the practical significance of fractional diffusion frameworks, where the Caputo-Hadamard operator captures memory effects critical for accurate prediction of contaminant plume evolution in heterogeneous aquifers that classical models significantly underestimate in terms of cleanup timeframes and barrier performance requirements. The research establishes this approach as the current state-of-the-art for fractional inverse source problems, providing essential tools for environmental engineering applications including remediation design, exposure assessment, and long-term monitoring strategies in complex groundwater systems.