An inverse problem of determining the orders of systems of fractional pseudo-differential equations

TL;DR

This paper addresses the inverse problem of uniquely determining the fractional orders in systems of pseudo-differential equations using Fourier transform data at a fixed time, crucial for modeling dynamical processes.

Contribution

It proves the unique recoverability of the fractional orders in linear systems of pseudo-differential equations from Fourier transform data at a fixed time.

Findings

Unique recovery of fractional orders established

Fourier transform at fixed time suffices for identification

Applicable to modeling of dynamical systems

Abstract

As it is known various dynamical processes can be modeled through the systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with determining the adequate orders of time-fractional derivatives in the sense of Riemann-Liouville or Caputo. This problem is qualified as an inverse problem. The right (vector) order can be found utilizing the available data. In this paper we consider this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution $\widehat{U}(t, \xi)$ evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of the system of governing pseudo-differential equations.