2022/10/28 by Ravshan Ashurov, Ashurov, Ravshan, Shakarova Marjona +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical methods in inverse problems #Primary 35R11 #Secondary 34A12
paper · pdf · doi:10.48550/arxiv.2211.00081
openalex publication_date 2022/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,t0)= ψ(x) is taken as the over-determination condition, where t0 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.