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Fractional boundary layer flow of Maxwell fluid on an unsteady stretching surface was investigated. Time–space dependent fractional derivatives are introduced into the constitutive equations of the fluid. We developed and solved the governing equations using explicit finite difference method and the L1-algorithm as well as shifted Grünwald–Letnikov formula. The effects of fractional parameters, relaxation parameter, Reynolds number, and unsteadiness parameter on the velocity behavior and characteristics of boundary layer thickness and skin friction were analyzed. Results obtained indicate that the behavior of boundary layer of viscoelastic fluid strongly depends on time–space fractional parameters. Increases of time fractional derivative parameter and relaxation parameter both cause a decrease of velocity while boundary layer thickness increase, but the space fractional derivative parameter and fractional Reynolds number have the opposite effects.