Abstract The memory-dependent derivative (MDD) is a new substitution for the fractional derivative (FD). It reflects the memory effect in a more distinct way. As an application, the representative heat diffusion process is remodeled with it. In fact, due to the existence of heat-conduction paradox, the time-space evolution mechanisms of this process are challenges to the modelers. The paradox cann’t be ascribed to the classical Fourier law, and the results show that the newly-constructed temporal MDD model is more reasonable than the Maxwell-Cattaneo, the temporal FD, the spatial FD and the common ones. Moreover, different mediums may accord with different memory times and weighted functions. This freedom of choice reflects the flexibility of MDD in modelling. It can be borrowed for exploring other diffusion problems.

中文翻译：

---

记忆依赖导数与分数导数（II）：重塑扩散过程

摘要 记忆相关导数（MDD）是分数导数（FD）的一种新替代方法。它以更鲜明的方式反映了记忆效应。作为一个应用程序，代表性的热扩散过程用它进行了改造。事实上，由于热传导悖论的存在，这一过程的时空演化机制对建模者来说是一个挑战。该悖论不能归因于经典傅里叶定律，结果表明，新构建的时间MDD模型比Maxwell-Cattaneo、时间FD、空间FD和普通模型更合理。而且，不同的媒体可能符合不同的记忆时间和加权函数。这种选择自由反映了 MDD 在建模方面的灵活性。可以借用它来探索其他扩散问题。