INITIAL-BOUNDARY-VALUE PROBLEMS FOR THE ONE-DIMENSIONAL TIME-FRACTIONAL DIFFUSION EQUATION

In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigen-functions of a certain Sturm-Liouville eigenvalue problem. For the one-dimensional time-fractional diffusion equation (D(t)(alpha)u)(t) = partial derivative/partial derivative x (p(x)partial derivative u/partial derivative x) - q(x) u + F(x, t), x is an element of (0, l), t is an element of (0, T) the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.