Abstract：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.
2016, Vol. 37 
