Solving inverse coefficient problems of non-uniform fractionally diffusive reactive material by a boundary functional method

Abstract

The inverse coefficient problems for estimating the unknown space-time dependent conductivity function and reaction coefficient function of a time-fractional diffusion reaction equation for non-uniform material are solved in the paper, without needing of initial condition, final time condition and internal measurement. We tackle these inverse coefficient problems supplementing with boundary data. After a homogenization technique by using these boundary data, a sequence of boundary functions are derived, which together with the zero element constitute a linear space. As an approximation, a boundary functional is proved in the linear space, of which the time-dependent energy is preserved for the energetic boundary function at each time step. The linear systems and iterative algorithms used to recover the unknown conductivity function and reaction coefficient with energetic boundary functions as bases are developed, which are convergent fast at each time step. The data required are merely the boundary temperatures and fluxes, and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the estimated results under large noise with exact solutions.