Identifying an unknown time-dependent boundary source in time-fractional diffusion equation with a non-local boundary condition

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

“…If α / ∈ (0, 1) or α ∈ (0, 1), but λ k = λ 0 for all k ≥ 1, then problem (2) has a unique solution and this solution has the form (39).…”

“…In this result, we do not need any monotonic premises on h(t) and/or the coefficients of A(t), which is the new aspect (and highlight) in this area of inverse problems. Many authors have considered an Equation (45) in which h(t) ≡ 1 and f (x) is unknown (see, e.g., [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]). Let us mention only some of these works.…”

“…The case of subdif-fusion equations, the elliptic part of A of which is an ordinary differential expression, is considered in [30][31][32][33][34][35][36]. The authors of the articles [37][38][39][40][41] studied subdiffusion equations in which the elliptic part of A is either a Laplace operator or a second-order operator. The paper [42] studied the inverse problem for the subdiffusion Equation ( 2) with the Cauchy condition.…”

On the Nonlocal Problems in Time for Time-fractional Subdiffusion Equations

“…Let us denote T k (0) = b k . Then the unique solution to the differential equation (23) with this initial condition has the form T k (t) = b k E ρ (−λ k t ρ ) (see, for example, [2], p.174, [3], [4], p. 17). From the nonlocal conditions of (23) we obtain the following equation to find the unknown numbers b k :…”

“…Finally, let 0 < α < 1 and λ k = λ 0 for k = k 0 , k 0 + 1, ..., k 0 + p 0 − 1, where p 0 is the multiplicity of the eigenvalue λ k 0 . Then the nonlocal problem (23) has a solution if the boundary function ψ(x) satisfies the following orthogonality conditions…”