On determining the fractional exponent of the subdiffusion equation

TL;DR

This paper reviews recent progress in identifying the fractional order in subdiffusion equations and introduces an asymptotic formula enabling the calculation of this order from solution data at a single point.

Contribution

It presents a novel asymptotic formula for determining the fractional derivative order in subdiffusion equations with negative eigenvalues.

Findings

Asymptotic formula for fractional order derived

Order can be calculated from solution at one point

Applicable when elliptic operator has negative eigenvalues

Abstract

Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists. A number of interesting results with a certain applied significance were obtained. This paper provides a short overview of the most interesting works in this direction. Next, we consider the problem of determining the order of the fractional derivative in the subdiffusion equation, provided that the elliptic operator included in this equation has at least one negative eigenvalue. An asymptotic formula is obtained according to which, knowing the solution at least at one point of the domain under consideration, the required order can be calculated.