Generalized time fractional diffusion problems solved by a Fourier method

We consider diffusion problems where the generalized fractional differential operator acting on the time variable extends the Caputo and Riemann–Liouville operators. Accordingly, two initial conditions are required. By implementing an elementary Fourier method, we have obtained existence and uniqueness for so-called mild problems. Finally, time-regularity of their solution is investigated without imposing any additional regularity assumption on the initial conditions. This issue is particularly challenging because of the “Riemann–Liouville part” of the fractional derivative.