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

Abstract

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