Identification of a time-dependent source term in a distributed-order time-fractional equation from a nonlocal integral observation

The ultra-slow diffusion process from internal source is governed by the inhomogeneous distributed-order time-fractional diffusion equation. We consider an inverse problem of recovering the time-dependent internal source from a nonlocal integral observation, which can be regarded as the spatial average of a pointwise observation. The uniqueness of the source reconstruction using observation data in a finite time interval is proven based on the energy estimate. Then, the reconstruction problem is reformulated as a minimization problem involving a Tikhonov regularizing term. By deriving the explicit representation of the gradient of the cost functional in terms of an adjoint problem, the conjugate gradient method is applied to obtain the numerical solution. Numerical results are presented to show the validity of the proposed algorithm. (C) 2019 Published by Elsevier Ltd.