Identification of a diffusion coefficient in a distributed order time fractional diffusion equation based on the conjugate gradient method

Abstract In this work, we consider an inverse problem of identify the space-dependent diffusion coefficient in the distributed-order time-fractional diffusion equation from boundary measurement data. A numerical algorithm is developed using the finite difference method to solve the direct problem, numerical results demonstrate the effectiveness and a certain degree of accuracy of this algorithm. To identify the diffusion coefficient, the problem is reformulated into a variational problem using the Tikhonov regularization method. The conjugate gradient method is utilized to tackle the variational problem by leveraging insights from a sensitivity problem and an adjoint problem. Numerical inversions are performed for diffusion coefficients with various functional forms, including scenarios with additional data influenced by random noise. Four numerical instances are presented to exhibit the efficacy and robustness of the method proposed in this paper.