Identification of the time-dependent source term in a multi-term time-fractional diffusion equation

The multi-term time-fractional diffusion equation is a useful tool in the modeling of complex systems. This paper aims to identifying a time-dependent source term in a multi-term time-fractional diffusion equation from the boundary Cauchy data. The regularity of the weak solution for the direct problem with homogeneous Neumann boundary condition is proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. On the other hand, the inverse time-dependent source term is formulated into a variational problem by the Tikhonov regularization, with the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate time-dependent source term. Numerical experiments for five examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.