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 α, 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 α. In particular, it is proved that for sufficiently large t the norm ||u(x, t)|| L 2 (Ω) of the solution decreases with respect to α. Moreover the inverse problem of determining the order of the derivative α is solved uniquely.Key words and phrases. The Rayleigh-Stokes problem, dependence of solution on the order of derivetive, inverse problem, determination of order of derivative, Fourier method.