  Time-fractional partial differential equations are nonlocal in time and show
an innate memory effect. In this work, we propose an augmented energy
functional which includes the history of the solution. Further, we prove the
equivalence of a time-fractional gradient flow problem to an integer-order one
based on our new energy. This equivalence guarantees the dissipating character
of the augmented energy. The state function of the integer-order gradient flow
acts on an extended domain similar to the Caffarelli-Silvestre extension for
the fractional Laplacian. Additionally, we apply a numerical scheme for solving
time-fractional gradient flows, which is based on kernel compressing methods.
We illustrate the behavior of the original and augmented energy in the case of
the Ginzburg-Landau energy functional.
