$(1+α)$阶分数阶微分方程的渐近积分
Asymptotic integration of $(1+α)$-order fractional differential equations
摘要：
在对泛函系数$a(t)$的一些简单限制下，建立了$(1+α)$ -阶分数阶微分方程${}_{0}^{\>i}{\cal O}_{t}^{1+α}x+a(t)x=0$，$t>0$解的长时间渐近公式，其中${}_{0}^{\>i}{\cal O}_{t}^{1+α}$是分数阶微分算子${}_{0}D_{t}^α(x^{\prime})$，$({}_{0}D_{t}^αx)^{\prime}={}_{0}D_{t}^{1+α}x$和${}_{0}D_{t}^α(tx^{\prime}-x)$之一.这里，${}_{0}D_{t}^α$表示$α\in(0,1)$阶的黎曼-刘维尔导数。对于给定的$a$，$b\in\mathbb{R}$，渐近公式读作$[a+O(1)]\cdot x_{\scriptstyle small}+b\cdot x_{\scriptstyle large}$ as $t\rightarrow+\infty$，其中$x_{\scriptstyle small}$和$x_{\scriptstyle large}$表示最终小解和最终大解，它们生成分数阶微分方程${}_{0}^{\>i}{\cal O}_{t}^{1+α}x=0$，$t>0$的解空间。
英文摘要：
\noindent{\bf Abstract} We establish the long-time asymptotic formula of solutions to the $(1+α)$--order fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x+a(t)x=0$, $t>0$, under some simple restrictions on the functional coefficient $a(t)$, where ${}_{0}^{\>i}{\cal O}_{t}^{1+α}$ is one of the fractional differential operators ${}_{0}D_{t}^α(x^{\prime})$, $({}_{0}D_{t}^αx)^{\prime}={}_{0}D_{t}^{1+α}x$ and ${}_{0}D_{t}^α(tx^{\prime}-x)$. Here, ${}_{0}D_{t}^α$ designates the Riemann-Liouville derivative of order $α\in(0,1)$. The asymptotic formula reads as $[a+O(1)]\cdot x_{\scriptstyle small}+b\cdot x_{\scriptstyle large}$ as $t\rightarrow+\infty$ for given $a$, $b\in\mathbb{R}$, where $x_{\scriptstyle small}$ and $x_{\scriptstyle large}$ represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation ${}_{0}^{\>i}{\cal O}_{t}^{1+α}x=0$, $t>0$.
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