Generalized Sommerfeld problem for time fractional diffusion equation: analytical and numerical approach

We consider analytical and numerical methods for solving the generalized Sommerfeld problem for time fractional diffusion equations (TFDE). This problem is to find the generalized diffusion coefficient lambda of TFDE and the order alpha of time derivative according to an additional information about a solution of TFDE. In the case of the steady-state anomalous diffusion process, the periodic boundary value problem without initial conditions for TFDE is solved and exact analytical solutions for inverse problems are obtained. When the initial phase of the process is considered, we use the modified Monte Carlo method to obtain the inverse problem data. In this case, inverse problems are formulated as residual function minimization problems, the Levenberg-Marquardt algorithm for residual function minimization is used and numerical results are presented.