On the regional gradient observability of time fractional diffusion processes

TL;DR

This paper investigates the regional gradient observability of time fractional diffusion systems, proposing methods to reconstruct initial gradients in subregions without initial data, applicable to anomalous sub-diffusion processes.

Contribution

It introduces the concept of regional gradient observability for Riemann-Liouville fractional diffusion systems and develops approaches for initial gradient reconstruction without initial state knowledge.

Findings

Characterization of sensors for regional gradient observability

Method for reconstructing initial gradients in subregions

Validation through application examples with various sensor types

Abstract

This paper for the first time addresses the concepts of regional gradient observability for the Riemann-Liouville time fractional order diffusion system in an interested subregion of the whole domain without the knowledge of the initial vector and its gradient. The Riemann-Liouville time fractional order diffusion system which replaces the first order time derivative of normal diffusion system by a Riemann-Liouville time fractional order derivative of order $\alpha\in (0,1]$ is used to well characterize those anomalous sub-diffusion processes. The characterizations of the strategic sensors when the system under consideration is regional gradient observability are explored. We then describe an approach leading to the reconstruction of the initial gradient in the considered subregion with zero residual gradient vector. At last, to illustrate the effectiveness of our results, we present several application examples where the sensors are zone, pointwise or filament ones.