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A uniqueness determination of the fractional exponents in a three-parameter fractional diffusion

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^βu(t,x)={\mathtt{L}_D^{α_1,α_2}} u(t,x), \ \ t\geq 0, \ x\in D, $$ where $\partial_t^β$ is the Caputo fractional derivative of order $β\in (0,1)$ and the differential operator ${\mathtt{L}_D^{α_1,α_2}}$ is the generator of a Lévy process, sum of two symmetric independent $α_1-$stable and $α_2-$stable processes and ${D}$ is the open unit interval in $\mathbb{R}$. We consider a nonlocal inverse problem and show that the fractional exponents $β$ and $α_i, \ i=1,2$ 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.