Inverse problem for the subdiffusion equation with fractional Caputo derivative

Abstract

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.

Uniqueness and Existence for Inverse Problem of Determining an Order of Time-Fractional Derivative of Subdiffusion Equation

“…Remark 2.7. In the paper [24], a lemma similar to the above lemma was proved for the diffusion and subdiffusion equations. In this paper g(t 0 ) = 0 and g(0) = 0 were for ρ = 1 and ρ ∈ (0, 1), respectively.…”

“…The problem of finding the function u(t) satisfying subdiffusion equation (1.1) with initial condition (1.2) is also called the forward problem. The forward problem is wellstudied in the literature, and the existence and uniqueness of the solution have been proved in various works, including [24], [26]. These works provide important theoretical foundations for studying the inverse problem.…”

“…As mentioned earlier, Theorem 3.1 for the diffusion equation (ρ = 1) with the additional condition u(x, t 0 ) = ψ has only been proven in the cases where Ω is an interval on R (see, [14]) or a rectangle in R 2 (see, [15]). The inverse problem (1.1)-(1.2) with the same additional condition, considering both the cases when the function g(t) changes sign and when it does not change sign, has been addressed in the work of Ashurov et al (see, [24]). However, the theorems we have presented above, for both the diffusion and subdiffusion equations, involve an integral additional condition (1.3).…”

“…Similar problems for the diffusion equation are studied in the well-known monographs of S.Kabanikhin [12] and the papers [13]- [19]. As for the subdiffusion equation, such inverse problems are studied in papers [20]- [24]. Let us mention some of the results obtained for the diffusion and subdiffusion equations.…”

“…In the paper [24] of the researchers analyzed subdiffusion equation with the Caputo derivative in which the Laplace operator forms the elliptic part. This paper focused on forward and inverse problems for the subdiffusion equation.…”