  In this article, we consider the space-time Fractional (nonlocal) diffusion
equation $$\partial_t^\beta u(t,x)={\mathtt{L}_D^{\alpha_1,\alpha_2}} u(t,x), \
\ t\geq 0, \ x\in D, $$ where $\partial_t^\beta$ is the Caputo fractional
derivative of order $\beta \in (0,1)$ and the differential operator
${\mathtt{L}_D^{\alpha_1,\alpha_2}}$ is the generator of a L\'evy process, sum
of two symmetric independent $\alpha_1-$stable and $\alpha_2-$stable processes
and ${D}$ is the open unit interval in $\mathbb{R}$. 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 many
fields, including physics and environmental engineering. We also discuss the
numerical approximation of the inverse problem as a nonlinear least-squares
problem and explore parameter sensitivity through numerical experiments.
