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.

翻译：时间不折的局部偏差方程式在时间上不是本地的, 并显示出一个内在的内存效果 。 在这项工作中, 我们提议增加能量功能, 包括解决方案的历史 。 此外, 我们证明时间不折的梯度流问题与基于我们新能量的整数顺序问题相等 。 这个等值保证了增能的消散特性 。 整数梯度流流的状态功能在类似于分数拉平方圆的 Caffarelli- Silvestre 扩展的扩展域上发生 。 此外, 我们用数字方法来解决时间不折数梯度流, 以内核压缩法为基础 。 我们用 Ginzburg- Landau 能源功能来说明原始和增加能量的行为 。