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

TL;DR

This paper investigates initial-boundary problems for the time-fractional diffusion equation in the quarter plane, analyzing solutions' behavior as the fractional order approaches 1, thus connecting fractional and classical heat equations.

Contribution

It provides solutions for three initial-boundary-value problems involving the Caputo fractional derivative and examines their limits as the fractional order approaches 1.

Findings

Solutions for fractional diffusion problems are obtained.

The limit of solutions as pproaches 1 recovers classical heat equation solutions.

Asymptotic behavior of Wright functions is utilized in the analysis.

Abstract

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.