Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

TL;DR

This paper establishes an equivalence between time-fractional and integer-order gradient flows using an augmented energy functional that captures memory effects, and demonstrates this with numerical schemes and energy behavior analysis.

Contribution

It introduces a novel augmented energy functional that reflects memory effects and proves the equivalence of fractional and integer-order gradient flows.

Findings

The augmented energy dissipates over time, reflecting the memory effect.

Numerical schemes based on kernel compressing effectively solve fractional gradient flows.

The energy behavior of fractional and integer flows is comparable in Ginzburg-Landau models.

Abstract

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.