It is well fact that the computational complexity of unsteady and viscoelastic nanofluid can be measured by mathematical modeling that allows the suitable performance for variations from specific to particular rheological parameters. This manuscript addresses the fractionalized mathematical modeling of an unsteady incompressible Maxwell transient free convection flow of nanofluids in the existence of radiation as well as damped thermal flux by using the Caputo time-fractional derivative. The integral transform technique is invoked on the fractionalized governing equations to find the exact solution. The numerical solution to the problem is also attained by utilizing Stehfest and Tzou algorithms. Finally, for the sake of graphical illustration, the comparative analysis for validating both algorithms on velocity and temperature profiles is performed numerically. The temperature and velocity curves are reduced by growing α for small time, and behavior is contrary for large time. By growing values of [Formula: see text] declines the velocity profiles. Velocity is linearly proportional to relaxation time λ, and the boundary layer difference is decreased by increasing the time. Our achieved solutions via two different numerical methods, that is, Stehfest and Tzou are equal.