An inverse problem for a multidimensional fractional diffusion equation

Abstract We prove that the coefficients a ij (x)$a_{ij}(x)$ , q(x), and the domain Ω of a multidimensional fractional diffusion equation can be recovered uniquely from measurements u(b,t)$u(b,t)$ , t∈(t 0 ,t 1 )${t\in (t_0,t_1)}$ , at an arbitrary single point b inside a bounded domain Ω⊂ℝ n ${\Omega \subset \mathbb {R}^n}$ . From the measurements we first recover infinitely many spectral data (λ m ,φ m (x))${(\lambda _m,\varphi _m(x))}$ of the elliptic operator associated with the fractional diffusion equation. Then, the coefficients a ij (x)$a_{ij}(x)$ , q(x) are found from linear algebraic systems of the form Ay=b${Ay = b}$ , where A is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunction φ 1 (x)$\varphi _1(x)$ .