Distributed-order time-fractional wave equations

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 guarantees 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.