Determining the order of time and spatial fractional derivative

We study the initial boundary value problem for the equation $D^\rho_t u(x,t)+ (-\triangle)^\sigma u(x,t)=0$ in the $N$-dimensional domain $\Omega$ with the homogeneous condition Dirichlet. The fractional derivative is taken in Caputo’s sense. The existence and uniqueness of a strong solution for an arbitrary initial function from $L_2(\Omega)$ is proved. Next, we studied the inverse problem of simultaneously determining the order $\rho$ and the degree of the Laplace operator $\sigma$. Additional conditions are found that guarantee both the existence and uniqueness of solutions to the inverse problem under consideration.