Necessary non-local conditions for a time-fractional diffusion-wave equation

TL;DR

This paper investigates the time-fractional diffusion-wave equation with Riemann-Liouville derivatives, introducing integral operators with Wright functions, establishing non-local boundary conditions, and proving unique solvability with explicit solutions.

Contribution

It introduces new integral operators involving Wright functions and derives necessary non-local boundary conditions for the fractional diffusion-wave equation.

Findings

Derived necessary non-local boundary conditions.

Proved unique solvability of the problem.

Obtained explicit solutions for the equation.

Abstract

In this paper the time-fractional diffusion-wave equation with Riemman-Liouville fractional derivative is studied. The integral operators with the Wright function in the kernel, associated with the studied equation are introdused and their properties are investigated. In terms of these operators the necessary non-local conditions binding traces of solution and its derivatives on the boundary of a rectangular domain are found. By using the limiting properties of the Wright function the necessary non-local conditions for wave equation are obtained. With the help of the mentioned integral operator's properties a unique solvability of the problem with Samarskii integral condition for the diffusion-wave equation is proved. The solution is obtained in explicit form.