Fractional wave equations

23 May 2019
Ljubica Oparnica

The classical wave equation is derived from the system of three equations: The equation of motion of a (one-dimensional) deformable body, the Hook law as a constitutive equation, and the  strain measure, and describes wave propagation in elastic media. 
Fractional wave equations describe wave phenomena when viscoelasticity of a material or non-local effects of a material comes into an account. For waves in viscoelastic media, instead of Hook's law, a constitutive equation for viscoelastic body,  for example, Fractional Zener model or distributed order model of viscoelastic body, is used. To consider non-local effects of a media, one may replace classical strain measure by non-local strain measure. There are other constitutive equations and other ways to describe non-local effects which will be discussed within the talk.  
The system of three equations subject to initial conditions, initial displacement and initial velocity, is equivalent to one single equation, called fractional wave equation. Using different models for constitutive equations, and non-local measures, different fractional wave equations are obtained. After derivation of such equations, existence and uniqueness of their solution in the spaces of distributions is proved by the use of Laplace and Fourier transforms as main tool. Plots of solutions are presented. For some of derived equations microlocal analysis of the solution is conducted. 

  • PDE CDT Lunchtime Seminar