论文标题
用分数Caputo衍生物的子扩散方程的逆问题
Inverse problem for the subdiffusion equation with fractional Caputo derivative
论文作者
论文摘要
考虑使用分数Caputo衍生物确定子扩散方程的右侧的反问题。等式的右侧具有$ f（x）g（t）$的表格，未知是函数$ f（x）$。条件$ u（x，t_0）=ψ（x）$被视为过度确定条件，其中$ t_0 $是考虑域的某个内部点，$ψ（x）$是给定功能。傅立叶方法证明了在功能上的某些条件下，$ g（t）$和$ψ（x）$存在逆问题的解决方案，并且是唯一的。给出了一个示例，显示了某些签名函数$ g（t）$的逆问题解决方案的唯一性。结果表明，对于此类函数$ g（t）$的逆问题解决方案，必须满足方程式的椭圆形部分的某些正交条件，并且必须满足。
The inverse problem of determining the right-hand side of the subdiffusion equation with the fractional Caputo derivative is considered. The right-hand side of the equation has the form $f(x)g(t)$ and the unknown is function $f(x)$. The condition $ u (x,t_0)= ψ(x) $ is taken as the over-determination condition, where $t_0$ is some interior point of the considering domain and $ψ(x) $ is a given function. It is proved by the Fourier method that under certain conditions on the functions $g(t)$ and $ψ(x) $ the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution of the inverse problem for some sign-changing functions $g(t)$. It is shown that for the existence of a solution to the inverse problem for such functions $g(t)$, certain orthogonality conditions for the given functions and some eigenfunctions of the elliptic part of the equation must be satisfied.
