Identification of the Random Source in the Stochastic Caputo-Hadamard Time-Fractional Diffusion Equation

This paper investigates the inverse problem of identifying the unknown source term in a stochastic Caputo-Hadamard time-fractional diffusion equation, where the source term consists of a deterministic function and a stochastic process. By utilizing the statistical properties of final value data (including its expectation and variance), we recover both the deterministic and random terms. To address the inherent ill-posedness of the problem, we apply the quasi-boundary regularization method and provide both a priori and a posteriori error estimates. Finally, numerical experiments in one and two dimensions are conducted to validate the feasibility and effectiveness of the proposed method.