Reconstruction of Time-Dependent Source Terms in Fractional Diffusion Equations Using Carleman Estimates and Optimization Algorithms
DOI:
https://doi.org/10.56714/bjrs.51.2.7Keywords:
Carleman estimates, fractional diffusion, inverse problems, source reconstructionAbstract
This paper investigates the inverse problem of reconstructing time-dependent source terms in time-fractional diffusion equations with Caputo derivatives, which are widely used to model anomalous subdiffusive processes. The objective is to recover the temporal behaviour of unknown source functions from final-time boundary measurements. The methodology integrates both analytical and computational steps. First, a new Carleman estimate is established, ensuring uniqueness and conditional stability of the inverse problem. Next, this theoretical guarantee is used to design a numerical inversion framework. The forward problem is discretized using finite-difference and spectral schemes, and the source term is reconstructed through a regularized Levenberg–Marquardt optimization algorithm. Numerical experiments are then performed on synthetic and perturbed datasets to evaluate the accuracy and stability of the method. Results demonstrate strong resilience to noise and precise recovery of source terms under various test conditions. To the best of our knowledge, this is the first work to apply Carleman estimates to inverse problems in time-fractional diffusion equations, providing both theoretical validation and numerical implementation. The approach is applicable in diverse fields, including geophysics, biomedical imaging, and environmental monitoring.
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