单侧Levy稳定分布的精确显式概率密度
Exact and explicit probability densities for one-sided Levy stable distributions
摘要：
本文研究了随机系统中具有重要意义的指数为α（0< α<1）的单侧重尾Levy稳定概率分布函数g_α（x）的反常扩散和分数阶动力学。给出了g_α（x），0 PDMATHABMMx < \infty的精确显式表达式，满足\int_{0}^{\infty} exp（-px）g_α（x）dx = exp（-p^α），p>0，对所有α= l/k < 1，k和l为正整数.我们重现了所有已知的结果由k \leq 4，并提出了许多新的精确解，k > 4，所有的已知函数表示。这将允许对α进行“微调”，以使g_α（x）适应给定的实验情况。
英文摘要：
We study functions g_α(x) which are one-sided, heavy-tailed Levy stable probability distributions of index α, 0< α<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_α(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_α(x) dx = exp(-p^α), p>0, for all α= l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of αin order to adapt g_α(x) to a given experimental situation.
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