论文标题
本地和非本地分数多孔培养基方程
Local and Non-local Fractional Porous Media Equations
论文作者
论文摘要
最近观察到，S \＆P500价格回报的概率分布可以通过$ Q $ -Gaussian分布来建模，其中各种阶段（弱，强大的超级扩散和正常扩散）通过不同的拟合参数（Phys Rev. E 99，062313，2019）分开。在这里，我们分析了多孔介质方程的分数扩展，并表明所有这些都以广义$ Q $ -Gaussian函数的方式允许解决方案。考虑了三种“分数化”：\ textit {local}，指的是空间和时间的分数衍生物均为局部的情况； \ textit {non-local}，其中空间和时间分数衍生物都是非本地的；和\ textit {混合}，其中一个导数是局部的，而另一个衍生物是非本地的。虽然，对于\ textit {local}和\ textit {non-local}案例，我们找到了$ q $ -Gaussian解决方案，但它们的免费参数数量有所不同。这与适合真实数据的质量有所不同。我们测试了S \＆P 500价格回报的结果，并发现本地和非本地方案比经典的多孔媒体方程更适合数据。
Recently it was observed that the probability distribution of the price return in S\&P500 can be modeled by $q$-Gaussian distributions, where various phases (weak, strong super diffusion and normal diffusion) are separated by different fitting parameters (Phys Rev. E 99, 062313, 2019). Here we analyze the fractional extensions of the porous media equation and show that all of them admit solutions in terms of generalized $q$-Gaussian functions. Three kinds of "fractionalization" are considered: \textit{local}, referring to the situation where the fractional derivatives for both space and time are local; \textit{non-local}, where both space and time fractional derivatives are non-local; and \textit{mixed}, where one derivative is local, and another is non-local. Although, for the \textit{local} and \textit{non-local} cases we find $q$-Gaussian solutions , they differ in the number of free parameters. This makes differences to the quality of fitting to the real data. We test the results for the S\&P 500 price return and found that the local and non-local schemes fit the data better than the classic porous media equation.
