复杂介质中扩散和耗散行为的分数阶导数唯象建模

复杂介质一般是多相混合物.与普通固体、液体和气体相比，其力学行为具有明显的记忆、路径依赖性特征，难以用一般的经典力学模型来描述，因而显得反常.从数学力学建模上看，整数阶导数的局部极限定义不适合描述这样的非局部力学行为.分数阶导数实质上是微分-积分算子，能精确地刻画力学行为的全局相关特征.而且分数阶模型具有明确的统计物理解释.20世纪末至今，复杂介质反常力学行为的分数阶导数模型由于具有参数少，且参数的物理意义明确等突出优点，开始引起广泛关注.该文从唯象建模的角度，综述了分数阶导数和分形导数在复杂介质的反常扩散和频率依赖能量耗散建模中的应用与发展.

Fractional Differential Phenomenological Modeling for Diffusion and Dissipation Behaviors of Complex Media

A complex medium is generally a multiphase mixture. Unlike the classical solid, liquid and gas, its mechanical behaviors exhibit anomalous features such as the memory and the path-dependence characteristics, which can hardly be well described with the classical mechanics models of integral-order derivatives. From the viewpoint of mathematical and physical modeling, the local limit definition of the integral-order derivative is not suitable to depict such non-local mechanical behaviors. The fractional derivative is essentially an integro-differential operator with underlying clear statistical physical explanation and can accurately describe the global correlation of complex mechanical behaviors. Since 1990s, the fractional derivative modeling for anomalous mechanical behaviors of complex media has attracted extensive attention due to its merits of fewer parameters with clear physical explanations. From the phenomenological modeling perspective, a review was made on the applications and developments of the fractional and fractal derivative models for the diffusion and energy dissipation behaviors of complex media.