Determining the order of time and spatial fractional derivatives

The paper considers the initial‐boundary value problem for equation Dtρu(x,t)+(−Δ)σu(x,t)=0,ρ∈(0,1),σ>0$$ {D}_t^{\rho }u\left(x,t\right)+{\left(-\Delta \right)}^{\sigma }u\left(x,t\right)=0,\rho \in \left(0,1\right),\sigma >0 $$, in an N‐dimensional domain Ω$$ \Omega $$ with a homogeneous Dirichlet condition. The fractional derivative is taken in the sense of Caputo. The main goal of the work is to solve the inverse problem of simultaneously determining two parameters: the order of the fractional derivative ρ$$ \rho $$ and the degree of the Laplace operator σ$$ \sigma $$. A new formulation and solution method for this inverse problem are proposed. It is proved that in the new formulation the solution to the inverse problem exists and is unique for an arbitrary initial function from the class L2(Ω)$$ {L}_2\left(\Omega \right) $$. Note that in previously known works, only the uniqueness of the solution to the inverse problem was proved and the initial function was required to be sufficiently smooth and non‐negative.