考虑时空相关的分数阶对流-弥散方程及其解

本文在考虑弥散过程的时空相关性的基础上，用非局域性的处理方法，将二阶对流-弥散方程进行推广得到了分数阶的对流-弥散方程，方程中弥散项和对时间的导数被分数阶导数所代替。此方程的柯西问题的格林函数解是一分数稳定分布密度函数。由方程的稳定分布密度函数解说明了局域等效弥散系数与弥散过程有关，得出了等效弥散系数与运移尺度有关，是运移距离的幂函数的结论。这一结论从理论上解释了弥散系数的尺度效应。最后，用一实验的实测数据对所得结果进行检验，检验结果很好地说明了弥散过程中的偏态特征和“拖尾”现象，而传统二阶对流-弥散方程的高斯分布解却不能解释。因此，用分数阶的对流-弥散方程比二阶对流-弥散方程能更好的描述溶质在多孔介质中的弥散行为。

The Fractional order advection-dispersion equation with temporal and spatial correlation and its solution

Based on the hypothesis that diffusion kernel behaves as an exponential function, we developed a non-local method with spatial and temporal correlation to introduce a fractional order advection-dispersion equation from the usually used local 2nd order advection-dispersion equation. The solution of this fractional order advection-dispersion equation is fractional stable probability distribution density with four parameters. Then the fractional stable probability distribution density is applied to explain the scaling property of the dispersion coefficient. Lastly, the fractional stable probability distribution density is applied to simulate the temple variation of the solute concentration in a test. The result conforms the test data very well. It explains the property of skewness and long-tail in the temple variation of the solute concentration. While the Gauss probability distribution has not these properties. So the result demonstrates that this fractional order advection-dispersion equation is better than the local 2nd order advection-dispersion equation to describe the movement of solute in porous medium.