In this chapter, we study the inverse problem of determining the time-depending reaction-diffusion coefficient in the Cauchy problem for the time-fractional diffusion equation with the Caputo operator and the Hilfer operator (generalized Riemann-Liouville fractional derivative). An overdetermination condition is specified a single observation at the point \(x=0\) of the fractional diffusion process. Here also new formulas have been proposed for representing the solution of the first (Dirichlet), second (Neumann), and third (Robin) initial-boundary value problems for one-dimensional fractional diffusion equations with the time-fractional Caputo derivative in half-line.