论文标题
关于两个子扩散方程的两个反问题解决方案的唯一性
On the uniqueness of solutions of two inverse problems for the subdiffusion equation
论文作者
论文摘要
让$ a $为任意的积极自助会操作员，在可分离的希尔伯特太空$ h $中定义。在非本地边界价值问题中确定方程右侧和功能$ ϕ $的逆问题$ d_t^ρu（t） + au（t）= f（t）= f（t）$（$ 0 <ρ<1，0 <ρ<1，0 <t \ t \ leq t $），$（之一）等式左侧的操作员$ d_t $表示Caputo衍生物。对于两个反问题，$ u（ξ_1）= v $被视为过度确定条件。证明了正在考虑的问题解决方案的存在和唯一定理。研究了常数$α$对问题解决方案的存在和独特性的影响。发现了一个有趣的效果：解决远期问题时，解决方案$ u（t）$的唯一性被违反，而当解决相同值$α$的逆问题时，解决方案$ u（t）$变得独特。
Let $A$ be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space $H$. The inverse problems of determining the right-hand side of the equation and the function $ϕ$ in the non-local boundary value problem $D_t^ρ u(t) + Au(t) = f(t)$ ($0 < ρ< 1, 0 < t \leq T$), $u(ξ) = αu(0) + ϕ$, ($α$ is a constant and $0 < ξ\leq T)$, is considered. Operator $D_t$ on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems $u(ξ_1) = V$ is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant $α$ on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution $u(t)$ was violated, while when solving the inverse problem for the same values of $α$, the solution $u(t)$ became unique.
