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 orderα∈(0, 1). Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function inR+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit whenα↗1of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems whenα= 1, and the fractional diffusion equation becomes the heat equation.