Distributed-order time-fractional wave equations

TL;DR

This paper studies distributed-order time-fractional wave equations in viscoelastic media, establishing conditions for existence, uniqueness, and analyzing solution properties, with implications for modeling wave propagation in complex materials.

Contribution

It introduces a measure-based approach for constitutive functions, derives thermodynamical restrictions, and proves well-posedness of the equations.

Findings

Existence and uniqueness of solutions are guaranteed under thermodynamical restrictions.

Support and regularity of the fundamental solution are characterized.

Wave velocities are discussed in the context of the model.

Abstract

Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the distributed-order constitutive law. In this work we propose to take positive Radon measures for the constitutive "functions". First, we derive a thermodynamical restriction on the constitutive measures which is easy to check, and therefore suitable for applications. Then we prove that the setting with measures in combination with the derived thermodynamical restriction guarantee existence and uniqueness of solutions for the distributed-order fractional wave equation. We further discuss the support and regularity of the fundamental solution, and conclude with a discussion on wave velocities.