局部结构导数及其应用

经典的导数建模方法刻画了特定物理量对时间或空间的变化率，较少直接考虑复杂系统介观时间-空间结构对其物理力学行为的重要影响。本文通过引入结构函数，提出了一种局部结构导数建模方法，以克服传统方法的不足。结构函数刻画了系统的时间-空间特征，实际上是一个时空变换，基于其上的结构导数能够描述复杂问题介观时空结构与特定物理量的因果关系，减少模型参数，降低计算成本。我们可通过问题的广义基本解或已知统计分布的概率密度函数，推导出其系统的结构函数。两类应用实例表明，基于对数结构函数的结构导数方法可以描述软物质中的特慢扩散现象，也可用来建立以Weibull分布的概率密度函数为结构函数的可靠性结构导数扩散方程。

LOCAL STRUCTURAL DERIVATIVE AND ITS APPLICATIONS

The classical derivative modelling approach describes the change rate of a certain physical variable with respect to time or space and considers to less extent the important influence of mesoscopic time-space fabric of a complex system on its physical behaviours. This report introduces the structural function and proposes the local structural derivative modeling approach to overcome the shortcoming of the traditional derivative approach. The structural function characterizes the time-space structures of system of interest and in fact is a time-space transform. By using the structural function, the structural derivative can describe causal relationship of mesoscopic time-space structure and certain physical behavior in a simple fashion and less computing costs. We can obtain the structural function by using the fundamental solution or the probability density function of statistical distribution. Two applications in this study show that the proposed structural derivative can well describe the ultraslow diffusion with the logarithmic function as its structural function in soft matters and derive the structural derivative diffusion equation of reliability using the structural function based on the probability density function of Weibull distribution.