Received: 1 July 2019 / Accepted: 1 August 2019 / Published: 10 August 2019
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.
Recently, 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 , charge transport in amorphous semiconductors  or heating processes of the heterogeneous rod . 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. Then, some model-based investigations are needed to deal with their rather complex dynamical behaviors.
Source seeking is a fundamental issue in nature and, currently, different approaches have been developed to study it for the non-fractional diffusion systems (see monographs [9,10] and the survey  for example). This is motivated by the fact that, in some practical applications, such as the pollution problems, the danger may increase if the source remains unknown . However, from a practical point of view, engineers are more interested in the sensing problem that, if a source is detectable, how can it be identified based on a limited number of sensor measurements. Then, in this paper, we consider this source sensing problem for the subdiffusion processes governed by time fractional diffusion systems.
Motivated by these above considerations, in this paper, we deal with the following time fractional diffusion systems with a Riemann–Liouville fractional order derivative:
where is an open bounded subset with a smooth boundary , and represent the Riemann–Liouville fractional order derivative and integral, respectively. Here, denotes the unknown source to be specified later and A is the infinitesimal generator of a strongly continuous semigroup in . It is supposed that is a self-adjoint uniformly elliptic operator and, in addition, , where V is a Hilbert space such that with continuous injections ( is the dual of V).
It is worth noting that, although the initial condition for Riemann–Liouville type time fractional diffusion system does not take the same form as that of non-fractional differential equations, expressions like in system (1) make sense. The reason is that it does not require a direct experimental evaluation of these fractional integrals. Instead, one can get it by measuring the initial values of its “inseparable twin”, which is obtained based on some basic physical law for the particular field of science. That is, the physical meaning for the Riemann–Liouville fractional integral of a function is equivalent to the initial value of its “inseparable twin”. For example, in the fractional Voigt model (a spring and a spring-pot in parallel) of viscoelasticity, the physical meaning of a Riemann–Liouville fractional integral of the unknown strain is in fact identical to the initial condition of its “inseparable twin”—the stress . This is also consistent with the known fact that the spring in the Voigt model only affects long-term behavior. For more “inseparable twins”, we refer the reader to e.g., monographs [14,15] for more information on the pair of current and voltage in electrical circuits or the pair of temperature difference and heat flux in heat conduction, etc.
The applications of system (1) are rich in the real world. As stated in , system (1) is usually used to describe the dynamic process in spatially inhomogeneous environments. Typical examples include the flow through porous media with a source or sensing the source of groundwater flow, etc. The corresponding sensing techniques cited in this paper can also be used to enable more complex tasks such as landmine clearing, the disease spreading control in agriculture lands or the crowd evacuation in the case of emergencies.
Let the limited number of sensor measurements be given by
where depends on the structure of sensors and Z is a Hilbert space. Then, the source sensing problem can be stated as follows:
Given the measurements , find a source S such that the solution of system (1) satisfies
Several questions arise in such problems: can the available measurements z uniquely determine S? If so, how does S depend on z and is there an approach to determine it (sensing)?
In the past two decades, several numerical algorithm approaches have been proposed for the source sensing problem of non-fractional diffusion systems. In , fast algorithms to solve the source sensing problem for elliptic partial differential equations (PDEs) were presented, in which the solution was approximated by using the Fourier–Galerkin truncated method. By using the multidimensional frequency estimation techniques, a new framework for solving the source sensing problems for systems governed by linear PDEs was presented in [18,19]. In addition, if the source is assumed to be a sum of a finite number of Dirac delta functions at unknown locations, numerical algorithms for the source identification problem of linear heat equations and time-dependent advection-diffusion systems with a nonlinear reaction were considered in [20,21], respectively. For an overview of the optimisation approaches for pollution source sensing in groundwater, we refer the reader to  and the references cited therein, although it was confirmed in [23,24] that the transport phenomena under the ground should be a subdiffusion process governed by time fractional diffusion systems.
However, the investigations for the source sensing problem of time fractional diffusion systems are still very limited. This is due to the fact that there is a need for further studies on the optimization variant theory and gradient theory of fractional order systems. As a result, the above optimisation numerical methods seem to be inapplicable for system (1). Furthermore, a source detection method has been proposed by El Jai and Afifi , in which the source is characterized by three parameters according to its properties. Here, we adopt these concepts and introduce the notion of regional detection of unknown sources, where we are interested in the sensing of unknown source only in a subregion of the whole domain. As it will be shown, the idea of regional detection can surely save energy resources. In addition, it is easier to be implemented even for some cases where we have a possibility to detect it in the whole domain.
After the introduction, the mathematical concepts of source and detection are given in the next section. The third section is focused on the regional strategic sensors, regional spy sensors and their relationships. In Section 4, an approach on solving the source sensing problem is presented. Two applications are worked out in the end.
2. Preliminary Results
The purpose of this section is to introduce the notions of sources, detection and some basic results to be used thereafter.
Let . The definition of a source S is as follows:
 A source S is characterized by a triplet , where
represents the support of source that varies in time t;
defines the intensity of source in x at time t;
denotes the support of g and represents the life duration of source
Here, the support , which describes the moving trajectory of the source, is usually determined by the evolution of some dynamic systems. With this, S is said to be a
moving pointwise source if is reduced to a single point of for all ;
moving zone source if is reduced to a region of for all ;
boundary source if , and, in this case, we can define the similar pointwise/zone boundary sources;
fixed source if is independent of t, which may be pointwise, zone or boundary.
In addition, it is worth noting that, when discussing the sensing problem, the pointwise fixed source defined as , is always used. In this case, is independent of t, which is used to describe a single point of .
2.2. Regional Detection
Since the detection of a source can be done by neglecting its life duration, we consider the source as a couple . Let the set of such sources be . One has
Here, represents the set of parts of and denotes the space of functions . With this, can be a vector space with convenient scalar product operations.
A source S is said to be detectable on I if the knowledge of system (1), together with the output function (2), is sufficient to guarantee that the operator
However, in many cases, it is impossible or too costly to reconstruct all parameters of a source. Let be a non-empty, not necessarily connected subregion of In what follows, we introduce the concepts of regional detection.
Assume that the source is located in such that Considering the subspace
and defining the operator we obtain the following definition.
A source S is called to be detectable on I if (1), (2) is sufficient to ensure that is injective.
Note that a source, which is detectable, is called to be detectable if
2.3. Some Basic Results
To obtain our results, in this part, we present some basic results on fractional calculus.
().The Riemann–Liouville fractional integral of order for a function y is given by
where represents the Euler gamma function defined by and the right side is pointwise defined on .
().The Riemann–Liouville fractional derivative of order for a function y is defined as
provided that the right side is pointwise defined on .
Consider system (1); without loss of generality, suppose that and when .
be the Laplace transforms of functions y and S. Based on
system (1) is equivalent to which yields
Then, if there exists a function such that its Laplace transform is
let , El-Borai has shown in [27,28] that the unique solution of system (1) satisfies
As pointed out in , a sensor can be described by a couple such that represents the support of the actuator and f denotes its spatial distribution. Then, to obtain our main results, it is supposed that the measurements are made by p sensors and the output function becomes
Here, is a Hilbert space endowed with the inner product ; p denotes the number of the sensors, is the support of the sensors and represents their spatial distributions. In this case, .
Sensors are said to be strategic if the system (17) is weakly observable.
For the self-adjoint uniformly elliptic operator with Dirichlet boundary conditions, i.e.,
we see that the spectrum of is composed of eigenvalues and counting according to the multiplicities . Then, there exists a sequence such that
is real for each , and is the eigenvalue of A with multiplicities such that
, which is the non-trivial solution of the problem:
is the eigenfunction corresponding to such that where is Kronecker delta function concentrated at the origin. In addition, we get that the sequence forms a complete and orthonormal basis in and any can be expressed by
Note that the above assumptions on operator A is general. For example, if , , then is a symmetric operator. Considering the Dirichlet boundary conditions , we get that , and, in addition, forms a complete and orthonormal basis in .
We are now ready to state the following result.
Define matrices as
where , and . Then, the sensors are strategic if and only if
It follows from Definition 7 that the sensors are strategic if and only if
is known as the generalized Mittag–Leffler function in three parameters. In particular, write and for short when and respectively. With this, we have
which is following from the property for some [29,30]. Consequently, the necessary and sufficient condition for strategic sensors is that
Finally, we cover our proof by using Reductio ad Absurdum.
Necessity. If there exists a non-zero vector satisfying Then, we can construct a non-zero element with , for which
This implies that the sensors are not strategic.
Sufficiency. If the sensors are not strategic, we can find a element , such that Since for all , there exists some satisfying
Consequently, if , it is sufficient to see that The proof is complete. □
3.2. Regional Spy Sensors
Consider system (1) with measurements given by p sensors , we state the following definition of regional spy sensors, which may lead to numerous problems and pose challenging research topics at the same time.
Sensors are said to be spy sensors if they can detect any unknown sources in .
3.3. The Relationships between Spy Sensors and Strategic Sensors
Note that the detection problem and the observation problem are different . Consequently, it leads immediately to the difference between strategic sensors and spy sensors.
Strategic (strategic) sensors are spy (spy) sensors, while the converse is not true.
Based on the conclusion in  that is injective but not surjective, it is not difficult to see that, if sensors are strategic, they are spy sensors, while the converse fails. Here, may be whole domain. The proof is finished. □
In addition, we explore the following further result. For the sake of convenience, it is assumed that in the following discussion by realizing that system (1) is linear.
Suppose that g in S satisfying . Then, are spy sensors if and only if they are strategic sensors.
From Lemma 1, strategic sensors are spy sensors. Then, we next focus on showing its converse.
For any unknown sources , define the operator as
Based on Definitions 3 and 8, the necessary and sufficient condition for the spy sensors is that is injective. Then, if the sensors are not strategic, by Theorem 1, there exists an element such that for some That is,
Therefore, since and let One has
where is the source having as its intensity. This means that is not detectable. As a result, we conclude that are not spy sensors and the proof is finished. □
4. Source Sensing Approach
In this section, we show how to identify the source under the hypothesis that are spy sensors.
If are spy sensors, then the source in system (1) can be uniquely identified by the observation z as the unique solution of the following equation
That is, given any , the equality could imply .
Let be the solution of system
Then, the difference satisfies
In what follows, we divide the proof into three steps.
Step 1, we consider the following semi-norm
and show that defines a norm for the space . For this, we only need to prove that any with could yield . Indeed, by Definitions 3 and 8, since are spy sensors, we get that is injective, i.e., could imply . With this, we conclude that is a Hilbert space endowed with the norm and the inner product
Step 2, we prove that the operator given by
is an isomorphism from space into its dual Here, denotes the adjoint operator of Indeed, given any , by (15), one has
Then, the duality relationship and (28) yield that
it leads to Therefore, is a continuous operator and has a unique extension to such that
Moreover, we obtain that the linear operator is continuous. Then, is an isomorphism from to , which is following from (45) and (47).
Step 3, based on Theorem 1.1 of , to complete the proof, we only need to show that is a coercive operator. That is, there exists a positive constant such that
In fact, with these above preliminaries, if is injective, is Hilbert space endowed with the norm and the inner product
For any one has
Then, (36) has a unique solution. This means that any satisfying the equality could yield . Consequently, the unknown source S is uniquely identified and the proof is finished. □
From Theorem 3, if are spy sensors, we get that the operator is injective by using the knowledge of the considered time fractional diffusion system and the sensor measurements. The main tool used in the above proof is the duality theory and our results could be used for sensing both the bounded time-varying space-dependence source (the zone source) and the unbounded time-varying space-dependence source (the pointwise source). With this, we see that the obtained results can be considered as a generalization of that in . In particular, if and in the integral (15) is uniformly bounded with respect to all , i.e., , holds for some constant , we have 
which is convergent and is consistent with the conditional stability results in Theorem 3.1 of .
Note that Theorem 3 is obtained by assuming that the measurement doesn’t contain noise and the considered domain is regular so that the eigenvalue pairing of operator A satisfying Equations (21) and (22) is obtained. However, these assumptions may fail in some practical applications. For this, due to the memory effect of the fractional derivative, more new properties on fractional derivatives (or on Mittag–Leffler functions) and more constraints on a system operator are required. While interesting, we conclude that source sensing problems for time fractional diffusion systems under uncertain measurements of an irregular bounded domain and their robust analysis are of great interest.
Next, we give a concrete algorithm to recover the unknown fixed zone source . Here, denotes the support of the source and represents its intensity.
Since forms a complete and orthonormal basis in , can be rewritten as follows:
Then, the source sensing problem is converted to identify the value of coefficients By (36), one has
With this, set and . For a big enough integer J, can then be approximated by solving the following equation:
It is worth mentioning that the matrix of (55) is positive. Consequently, we have
5. Further Remarks
Realize that the Caputo fractional order derivative is another widely used derivative in fractional order systems; in this section, we consider the source sensing problem for the following time fractional diffusion system with a Caputo fractional derivative:
where denotes the Caputo fractional derivative.
Taking a Laplace transform on both sides of system (57), it yields that
For the approach on identifying the source governed by system (57), however, the conclusions obtained in previous sections will never hold if the measurements are defined as in (19). This is due to the fact that is usually not equal to if . Then, some new revised definition of the measurements should be introduced.
Observing that for any , following from (59), if the sensor measurements are revised to be given by p sensors as follows:
Consider system (57) with , define , and we obtain the following result.
System (57), (61) with is said weakly observable if
Define matrices as
where , and . Then, the sensors are strategic for system (57) with if and only if
Since the proof of Theorem 4 is very similar to that of Theorem 1, we omit it.
Consider system (57), let the operator be given by
The source sensing problem is stated as follows:
Given the measurements defined by (61), find a source S such that the solution of system (57) satisfies
Assume that are spy sensors. Then, the source in system (57) can be uniquely identified by the observation z in (61) as the unique solution of the following equation
For any two solutions , of system (57), their difference satisfies
Then, we divide the proof into three steps.
Step 1, if are spy sensors, we get that is injective. Then, the semi-norm
defines a norm for the space . Therefore, is a Hilbert space endowed with the norm and the inner product
Step 2, given any , since
The duality relationship leads to
Similar to Step 2 of Theorem 3, we get that the operator is an isomorphism from to its dual
Step 3, for any one has
Then, Theorem 1.1 of  yields that (67) has a unique solution. As a result, the unknown source S in system (57) is uniquely identified by the observation (61). The proof is finished. □
Similarly, for any unknown fixed zone source , by (52) and (67), we have
For big enough integer J, then can be approximated by multiplying both sides of (74) with as follows:
where and . With this, we obtain that
6. Numerical Examples
The aim of this numerical work is to identify a fixed zone source according to the methods given in Section 4.
Let be an open bounded subset with smooth boundary , we consider the following system
where denotes the Laplace operator and represents the unknown source to be sought. The measurements are made by p sensors as follows:
Based on the arguments in Section 4, the sensing problem of system (77) under measurements (78) can be solved via the following applicable steps:
Initial data , T and the spy sensors ;
Given big enough integer J, obtain and for all ;
Solve the problem (55) to get and then obtain g based on (52).
6.1. One-Dimensional Case
Let . We get that , and . Here, forms a complete and orthonormal basis in . Consider a fixed zone source with the intensity g given by
where denotes the support of the source. Suppose that the measurements are made by one sensor . By Theorem 1, we obtain
The sensor is strategic for some subregion if and only if
holds true for all
Since , it then follows that
Theorem 1 shows that (80) holds and the proof is complete. □
In particular, if the support D of sensor reduces to a point , we see that (80) is equal to Without loss of generality, let the measurements be given by a pointwise sensor located at with the unit spatial distribution. Then, sensor reduces to . With this, we get that
which is injective. Therefore, and
Moreover, one has
Let . Figure 1 shows how the approximated is close to g.
6.2. The Cases of
This part focuses on the system (77) in . In this case, the eigenvalues and corresponding eigenfunctions of operator ▵ are and , respectively. Here, Let the system (77) be excited by a fixed zone source with such that
Then, the support of S is and its presentation is given by of Figure 2.
Suppose that the measurements are made by one sensor . It follows that
The sensor is strategic for subregion if and only if for all
The proof can be easily obtained similar to the proof of Proposition 1 following from Theorem 1. Then, we omit it.
Let and . Then, the sensor reduces to , which is a spy sensor. Moreover, it follows that is injective and is defined as
Therefore, Theorem 3 shows that the source S can be sought from observation z by solving the equation . Let . We get that
Then, we refer the reader to and of Figure 2 on how close is the approximated to the exact g when .
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. The characterizations of regional strategic sensors, regional spy sensors and their relationships are presented. We discuss an approach on how to identify the unknown source only in a subregion of the whole domain by using the HUM. 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. The results here can be regarded as an extension of the results in . Moreover, we claim that some regularization method such as an iterative regularization method in  can be introduced to combine the concrete algorithm in this paper. Therefore, the source sensing problems for a fractional order distributed parameter systems by combining a regional detection method and the iterative regularization method as well as their comparisons with existing methods are of great interest.
All authors contributed to this research. F.G. designed the research and wrote the paper. W.S. contributed to the numerical simulation and assisted in writing the paper. Y.C. contributed to designing the research and revised the manuscript.
This research was funded by the National Natural Science Foundation of China (NSFC) Grant No. 41801365 and the Fundamental Research Funds for the Central Universities, China University of Geosciences, Wuhan Grant No. CUGGC05 and the Hubei NSFC Grant No. 2019CFB158.
Conflicts of Interest
The authors declare no conflict of interest.
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The exact intensity g and approximate intensity of the unknown fixed source.
The exact intensity g and approximate intensity of the unknown fixed source.
Comparison between the exact intensity g and the approximate intensity .
Comparison between the exact intensity g and the approximate intensity .