INVERSE PROBLEM FOR SUBDIFFUSION EQUATION WITH THE INTEGRAL OVER-DETERMINATION CONDITION

In this paper, we study the inverse problem of determining the right-hand side of the subdiffusion (and diffusion) equation with a fractional Riemann-Liouville derivative. The right-hand side of the equation has the form g(t)f(x), where the function f(x) is unknown. As an over-determination condition, we adopt the integral condition \(\int \limits _0^Tu(x,t)dt=\psi (x)\) $\underset{0}{\overset{T}{\int }}u(x,t)dt=\psi (x)$ . Here, \(\psi (x)\) $\psi (x)$ is a given continuous function. A criterion for the uniqueness of a solution to the inverse problem is found. For a sign-preserving function g(t), the existence and uniqueness of a solution are proved. If the function g(t) changes sign, then necessary and sufficient conditions for the existence of a solution are found. But in this case, the solution may not be unique. All the presented results are new for diffusion equations as well.