Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid

Abstract

This paper derives analytical solutions for a class of new multi-term fractional-order partial differential equations, which include the terms for spatial diffusion, time-fractional diffusion (multi-term) and reaction. These models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic Oldroyd-B fluid and Burgers fluid. By using a modified separation of variables method, the governing fractional-order partial differential equations are transformed into a set of fractional-order ordinary differential equations. Mikusiński-type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients. The solutions are expressed in terms of multivariate Mittag-Leffler functions. Different situations for the unsteady flows of generalized Oldroyd-B fluid and Burgers fluid due to a moving plate are considered via examples. Integral representations of the solutions are presented. It is shown that the presented results reduce to the corresponding results for classical Navier–Stokes, Oldroyd-B, Maxwell and second-grade fluids as special cases.