Abstract
Motivated by the fact that the danger may increase if the source of pollution problem remains unknown, in this paper, we study the source sensing problem for subdiffusion processes governed by time fractional diffusion systems based on a limited number of sensor measurements. For this, we first give some preliminary notions such as source, detection and regional spy sensors, etc. Secondly, we investigate the characterizations of regional strategic sensors and regional spy sensors. A regional detection approach on how to solve the source sensing problem of the considered system is then presented by using the Hilbert uniqueness method (HUM). This is to identify the unknown source only in a subregion of the whole domain, which is easier to be implemented and could save a lot of energy resources. Numerical examples are finally included to test our results.
Highlights
The studies of transport dynamics in complex systems which exhibit the subdiffusion property have attracted increasing attention
The aim of this paper is to discuss the source sensing problem in a subdiffusion process by a regional detection method motivated by the great potential applications in environmental problems
Some comparison results are given between time fractional diffusion system with a Riemann–Liouville fractional order derivative and that with a Caputo fractional order derivative
Summary
The studies of transport dynamics in complex systems which exhibit the subdiffusion property have attracted increasing attention. Typical examples include the water in membranes for fuel cells [1], charge transport in amorphous semiconductors [2] or heating processes of the heterogeneous rod [3]. It is worth mentioning that the mean squared displacement of subdiffusion process is a power-law function of fractional exponent, which is smaller than that of the Gaussian diffusion process [4,5]. Due to the strong interactions between components in these processes, a rather complex dynamical behavior would emerge. Note that a fractional order derivative itself is a kind of convolution and naturally links to subdiffusion processes, time fractional diffusion system is confirmed in [5,6,7,8] to be used to efficiently describe these subdiffusion processes. Some model-based investigations are needed to deal with their rather complex dynamical behaviors
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