Non-local time problem for the Rayleigh--Stokes type fractional equations

TL;DR

This paper investigates the Caputo fractional derivative version of the Rayleigh--Stokes problem, providing explicit solutions and analyzing existence, uniqueness, and regularity, filling a gap in the mathematical understanding of fractional fluid dynamics equations.

Contribution

It offers the first explicit analytical solution for the Caputo derivative case and a systematic study of well-posedness for these fractional equations.

Findings

Explicit analytical solution derived for Caputo fractional Rayleigh--Stokes problem.

Proved existence and uniqueness of solutions.

Analyzed regularity properties and non-local problem behavior.

Abstract

Despite the growing interest in fractional generalizations of classical fluid dynamics equations, the fractional Rayleigh--Stokes problem has previously been studied almost exclusively using the Riemann--Liouville fractional derivative. To the authors' knowledge, an explicit analytical form of the solution for the Caputo derivative case has not been established in the literature, and before this work, no systematic study of the existence, uniqueness, or regularity properties of this formulation has been conducted. In this paper, we fill this gap by considering the Rayleigh--Stokes equation with the Caputo fractional time derivative of order $\rho \in (0, \, 1)$. Using the Laplace transform and Fourier methods, as well as special functions, we perform a rigorous well-posedness analysis of the corresponding initial boundary-value, non-local, and backward problems.