On the explicit formula linking a function to the order of its fractional derivative

In this paper, given a certain regularity of a function $v$, we derive an explicit formula relating the order $\nu_0\in(0,1)$ of the leading fractional derivative in a fractional differential operator $\mathbf{D_t}$ with the variable coefficients $r_i=r_i(x,t)$ and the function $v$ on which this operator acts. Moreover, we discuss application of this result in the reconstruction of the memory order of semilinear subdiffusion with memory terms. To achieve this aim, we analyze some inverse problems to multi-term fractional in time ordinary and partial differential equations with smooth local or nonlocal additional measurements for small time. In conclusion, we discuss how this formula may be exploited to numerical computation of $\nu_0$ in the case of discrete noisy observation in the corresponding inverse problems. Our theoretical results along with the computational algorithm are supplemented by numerical tests.