Summary: The viscoelastic fluid exhibits significant elastic solid and viscous liquid responses which do not rise in Newtonian fluid because flow of viscoelastic fluid has the Markovian property. In order to describe the Markovian verses non-Markovian properties, the dual definitions of fractional differentiations have been invoked on the Oldroyd-B fluid model. The Oldroyd-B fluid model is saturated by porous medium and magnet subject to no slip assumptions on the governing equations of flow. The fractional differential operators of Atangana-Baleanu and Caputo-Fabrizio are imposed on the governing equations then solved by Fourier sine and Laplace transforms. The solutions are expressed into compact form by means of elementary functions, infinite series, theorem of convolution and generalized special function so called M function. The velocity field obtained via two types of fractional approaches is depicted for disclosing the hidden impact of relaxation and retardation times on Markovian and non-Markovian properties of viscoelastic fluid. Finally, our feasible analysis resulted that flow of viscoelastic fluid is dependent on its present state which leads to Markovian and non-Markovian dynamics.
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