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

TL;DR

This paper derives an explicit formula linking the order of a fractional derivative to the coefficients and function, with applications in reconstructing memory order in subdiffusion models and solving inverse problems.

Contribution

It introduces a new explicit formula connecting fractional derivative order with coefficients and functions, aiding in inverse problem solutions and numerical reconstruction.

Findings

Derived an explicit formula for fractional derivative order.

Applied the formula to reconstruct memory order in subdiffusion.

Provided numerical tests demonstrating the approach.

Abstract

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.