On the Initial-Boundary Problem for the Time-Fractional Diffusion Equation in the Quarter Plane

Taking into account the asymptotic behavior of some Wright functions and the existence of bounds for the Mainardi and the Wright function $W(-x,\frac{\alpha}{2}, 1)$ in $\mathbb{R}^+$ , three different initial-boundary-value problems for the time-fractional diffusion equation in the quarter plane, where the time-fractional derivative is taken in the Caputo sense of order $\alpha$ $\in (0,1)$ are solved. Moreover, the limit when $\alpha \nearrow 1$ of the respective solutions are analyzed, recovering the respective solutions of the classical boundary-value problems when $\alpha=1$ and the fractional diffusion equation becomes the heat equation.