Inverse problem for a three-parameter space-time fractional diffusion equation

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^\beta u(t,x)=\mathcal{L} u(t,x), \ \ t\geq 0, \ -1stable processes. We consider a nonlocal inverse problem and show that the fractional exponents $\beta$ and $\alpha_i, \ i=1,2$ are determined uniquely by the data $u(t; 0) = g(t), 0 < t < T.$ The uniqueness result is a theoretical background for determining experimentally the order of many anomalous diffusion phenomena, which are important in physics and in environmental engineering.