Distributed-order time-fractional wave equations

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.

References in corpus (4)

• Distributional framework for solving fractional differential equations
• Distributed order fractional constitutive stress-strain relation in wave propagation modeling
• Energy dissipation for hereditary and energy conservation for non-local fractional wave equations
• Micro-local and qualitative analysis of the fractional Zener wave equation

Cited by in corpus (3)

• Energy balance for fractional anti-Zener and Zener models in terms of relaxation modulus and creep compliance
• Wave propagation in three-dimensional fractional viscoelastic infinite solid body
• Stress and power as a response to harmonic excitation of a fractional anti-Zener and Zener type viscoelastic body