On the Initial-Boundary-Value Problem for the Time-Fractional Diffusion Equation on the Real Positive Semiaxis

We consider the time-fractional derivative in the Caputo sense of order alpha is an element of (0,1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when alpha NE arrow 1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when alpha = 1, and the fractional diffusion equation becomes the heat equation.