Inverse Problems for the Fractional Diffusion Equation with the Hilfer Operator

The article discusses inverse problems for the fractional diffusion equation with the Hilfer operator in time. The direct problem is the initial-boundary value problem for this equation with Cauchy-type initial data and Dirichlet boundary conditions. The first inverse problem, which involves determining a time-dependent coefficient, is reduced to an equivalent Volterra-type integral equation. The existence and uniqueness of the solution are proven using the contraction mapping principle. The second inverse problem involves determining a function dependent on the spatial variable on the right-hand side of the equation. This problem is studied using the Fourier method and the properties of the Mittag–Leffler function. The solution is constructed in the form of a series based on eigenfunctions.