论文标题
与分数Caputo时间衍生物相对于建模从普通次扩散到超扩散的过渡的另一个功能
Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion
论文作者
论文摘要
对于另一个功能$ g $（$ g $ - subdiffusion方程），我们使用带有分数CAPUTO时间衍生物的细分延伸方程来描述从普通次扩散到超级扩散的平稳过渡。等式用``普通''分数Caputo时间衍生物，超扩散来描述了普通的细胞扩散。我们发现，解决方案（Green的功能，GF）的功能$ G $ - 宽扩散方程以较小的时间的极限为GF的形式，用于长时间的限制。为了求解$ g $ - 量扩散方程，我们使用$ g $ - 拉普拉斯变换方法。结果表明，在长期限制中，GF的缩放属性和超级扩散的缩放属性和超级延伸的GF是相同的。我们得出的结论是，尽管对流程的随机解释有所不同，但在足够长的时间里，$ g $ - 量扩散方程式很好地描述了超扩散。然后，自相矛盾的是，具有分数时间衍生物的细节扩散方程描述了超延伸。在这里不是通过通过扩散的粒子进行异常跳远来实现的超级效果，而是通过大大增加粒子跳跃频率，而粒子跳跃频率是通过$ g $ - 连续的时间随机行走模型得出的。 $ g $ - 细节扩散方程被证明是相当一般的，它可用于建模过程，其中一种扩散随着时间的流逝而不断变化。此外，在对普通细胞扩散过程进行建模的一些方法（例如，在薄薄的薄膜下局部边界条件的推导，也可以用于模拟$ g $ - 亚扩散过程，即使该过程被解释为超扩散。
We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$--subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the ``ordinary'' fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz type spatial derivative. We find the function $g$ for which the solution (Green's function, GF) to the $g$--subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the $g$--subdiffusion equation we use the $g$--Laplace transform method. It is shown that the scaling properties of the GF for $g$--subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the $g$--subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the $g$--continuous time random walk model. The $g$--subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model $g$--subdiffusion processes, even if this process is interpreted as superdiffusion.
