Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation

In this paper, some uniqueness and existence results for the solutions of the initial-boundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain G×(0,T),G⊂Rn are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the Caputo–Dzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.