论文标题
$ \ mathbb {r}^n $中的亚扩散方程的初始边界值和反问题
Initial-boundary value and inverse problems for subdiffusion equations in $\mathbb{R}^N$
论文作者
论文摘要
考虑到具有椭圆运算符$ a（d）$ in $ \ mathbb {r}^n $的子扩散方程的初始值问题。傅立叶方法证明了该问题解决方案的存在和唯一定理。考虑到Caputo时间段衍生物的顺序为未知参数，研究了确定此顺序的相应逆问题。事实证明，解决方案$ \ hat {u}（ξ，t）$在固定时间实例上的傅立叶变换可独特地恢复未知参数。此外，在操作员$ a（d）$被其功率$ a^σ$取代的情况下，研究了类似的初始价值问题。最后，证明了确定分数衍生物相对于时间和程度$σ$的逆问题的解决方案的存在和唯一定理。我们还注意到，在解决逆问题时，已经证明了Mettag-Leffler函数的参数$ρ$ $e_ρ$的减少。
An initial-boundary value problem for a subdiffusion equation with an elliptic operator $A(D)$ in $\mathbb{R}^N$ is considered. The existence and uniqueness theorems for a solution of this problem are proved by the Fourier method. Considering the order of the Caputo time-fractional derivative as an unknown parameter, the corresponding inverse problem of determining this order is studied. It is proved, that the Fourier transform of the solution $\hat{u}(ξ, t)$ at a fixed time instance recovers uniquely the unknown parameter. Further, a similar initial-boundary value problem is investigated in the case when operator $A(D)$ is replaced by its power $A^σ$. Finally, the existence and uniqueness theorems for a solution of the inverse problem of determining both the orders of fractional derivatives with respect to time and the degree $ σ$ are proved. We also note that when solving the inverse problems, a decrease in the parameter $ρ$ of the Mettag-Leffler functions $E_ρ$ has been proved.
