Numerical reconstructions of a source term in a mobile-immobile diffusion model from the partial interior observation

We consider an inverse source problem in the two-time-scale mobile-immobile fractional diffusion model from partial interior observation. Theoretically, we combine the fractional Duhamel's principle with the weak vanishing property to establish the uniqueness of this inverse problem. Numerically, we adopt an optimal control approach for determining the source term. A coupled forward-backward system of equations is derived using the first-order optimality condition. Finally, we construct a finite element conjugate gradient algorithm for the numerical inversion of the source term. Several experiments are presented to show the utility of the method.