Inverse problem of determining the order of the fractional derivative in the Rayleigh-Stokes equation

Abstract

In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional derivative of order $\alpha$, which is used to describe the viscoelastic behavior of the flow. In this paper, we study the behavior of the solution of such equations depending on the parameter $\alpha$. In particular, it is proved that for sufficiently large $t$ the norm $||u(x,t)||_{L_2(\Omega)}$ of the solution decreases with respect to $\alpha$. Moreover the inverse problem of determining the order of the derivative $\alpha$ is solved uniquely.