The majority of researches for Falkner-Skan flow are dependent on the classical constitutive relations of viscoelastic fluids. Fractional Maxwell fluid model is introduced to depict Falkner-Skan flow innovatively in the paper. Moreover, the momentum equation studying the effects of buoyancy force is established. Analogy to constitutive relevance for fractional Maxwell fluid, fractional derivative is brought in Fourier's law and Fick's law. It is worth mentioning that heat source and chemical reaction are discussed. Finite difference method integrated with L1-algorithm is utilized to address the fractional governing equations, whose convergence is verified by constructing an example with accurate solution. Furthermore, the influences of pertinent physical parameters are analyzed diagrammatically. A fascinating phenomenon is discovered that all the velocity profiles initially rise to a maximal value due to the impact of buoyancy force and then decrease to the free flow velocity. Besides, the temperature and concentration distributions first increase slightly while decline significantly, which indicates the thermal relaxation and mass relaxation characteristic of Maxwell fluid.

The aim of the article is to research the unsteady magnetohydrodynamic stagnation-point flow of fractional Oldroyd-B fluid over a stretched sheet. According to the distribution characteristics of pressure and magnetic field near the stagnation point, the momentum equation based on fractional Oldroyd-B constitutive model is derived. Moreover, the modified fractional Fourier's law considering thermal relaxation-retardation time is proposed, which applies in both the energy equation and the boundary condition of convective heat transfer. New finite difference scheme combined with L1 algorithm is established to solve the governing equations, whose convergence is confirmed by constructing the exact solution. The results indicate that the larger relaxation parameters of velocity block the flow, yet the retardation parameters of velocity show the opposite trend. It is particularly worth mentioning that all the temperature profiles first go up slightly to a maximal value and then descend markedly, which presents the thermal retardation characteristic of Oldroyd-B fluid. Additionally, under the effects of temperature's retardation and relaxation parameters, the intersection of the profiles far away from stretching sheet demonstrates the thermal memory characteristic.