Abstract: The fractional calculus is used in the constitutiveequation of Maxwell viscoelastic fluid. An unsteady flowof viscoelastic fluid between two parallel plates is studied. For afractional derivative model of the generalized Maxwell viscoelasticfluid, the special equation of motion with fractional-order timederivatives is obtained. Additionally, the exact solution of the flowis obtained by using the theory of Laplace transform for fractionalcalculus.