1 Introduction
Many diffusion processes in nature and engineering, such as contaminants transport in the soil, oil flow in porous media, long distance transport of pollutants in the groundwater, etc., are referred to as anomalous diffusion, where the particle plume spreads slower or faster than predicted by the classical integerorder diffusion model. In recent three decades, fractional diffusion equations have been found to be efficient mathematical models to describe some anomalous diffusion phenomena (see [1, 6, 7, 17, 35, 43, 48], e.g.). The TFDE model, that is obtained from the classical diffusion equation by replacing the firstorder time derivative by a fractional derivative, is given as
where with , is an open bounded domain with smooth boundary, and denotes the state variable at space point and time , is the diffusion coefficient, is the fractional order, and the fractional derivative is defined by Caputo’s definition:
See, e.g., Podlubny [38] and Kilbas et al [24] for the definitions and properties of the fractional derivatives.
There are quite a lot of researches on equation (1.1) from numerics, and some works from theoretical aspects, we refer to [8, 11, 12, 13, 16, 33, 34, 36, 39] for the theoretical analysis on the solution of the forward problem, and also see the monographs [38], [24], [25] and [14]. On the other hand, we refer to [2, 10, 15, 21, 22, 29, 31, 32, 37, 42, 44] for the researches on inverse problems arising from the TFDEs. As we know the fractional order in a fractional model is a key parameter to characterize the heavytail subdiffusion with memory effect. However, it is always unknown for reallife problems which leading to inverse fractional order problems. The inverse problems of identifying parameters including fractional orders have been studied during the last decade. We refer to Alimor and Ashurov [3], Ashurov and Umarov [4], Ashurov and Zunnunov [5], Cheng et al [9], Hatano et al [18], Janno [19], Janno and Kinash [20], Jin and Kian [23], Li et al [26], Li et al [30], Li and Yamamoto [28], Sun et al [40], Tatar and Ulusoy [41], Yamamoto [45, 46], Yu et al [47], and so on.
It is noted that in the existing work on inverse fractional order problems, most of them were studied by using subdomain measurements or onepoint measurements at , or using subboundary data also at for arbiutrary given . An interesting problem is can we determine the fractional orders in theory only using limited discretized data? Exactly speaking, there is one order in equation (1.1) which is unknown, can we determine it uniquely only using one measurement?
It is difficult to give an answer in theory for the above question, but the situation could be changed if having suitable conditions. Here we are concerned with the inverse problem of determining the fractional order
in Eq. (1.1) using one measurement at one spacetime point. By the eigenfunction expansion method, the solution of the forward problem is expressed by the MittagLellfer function, and the inverse problem is transformed to a nonlinear algebraic equation. By choosing suitable initial values and the measured point, the nonlinear equation can be solved uniquely by the monotonicity of the MittagLellfer function. The unique solvability of the inverse problem is testified by theoretical examples.
The rest of the paper is organized as follows.
In Section 2, some preliminaries on the MittagLellfer function, and solvability on a nonlinear algebraic equation are introduced. The forward problem and its solution, and the inverse problem of identifying the fractional order using one measurement are given in Section 3. In Section 4, a local wellposedness of the inverse problem is obtained by the theory of solution to the nonlinear algebraic equation, and a conditional uniqueness is proved by suitably choosing the initial value and the measured time, and numerical testifications are presented. Conclusions are given in Section 5.
2 Preliminaries
We give some basic facts on the Gamma function, the MittagLeffler function and their properties, and the theory of solving a nonlinear algebraic equation. We refer to the monographs [14] and [27] for details.
2.1 The Gamma function and MittagLeffler function
The Gamma function is an analytic continuation of the factorial function in the entire complex plane, which is defined by
with . For the Gamma function there hold the formula , and the Stirling’s approximate formula
and the derivative’s formula
where denotes the derivative of the Gamma function, and is the Euler constant. On the Gamma function, we have the following assertion.
Lemma 1
For the parameter , there holds
Proof By (2.2) we have for large
Thanks to , we get
Following MittagLeffler’s classical definition, the oneparametric MittagLeffler function is defined by the power series
Obviously there is as , i.e., is a generalization of the exponential function .
On analytical properties of the MittagLeffler functions, we have the following assertions.
Lemma 2
(i) There exists a constant such that , .
(ii) For real number , the function is strictly monotone on , and decreasing on .
Lemma 3
Let for and given . There holds
where according to (2.3).
Proof By (2.7), it is easy to deduce that the derivative () is expressed by (2.8), where is given by (2.3), and we only need to prove the convergence of the series at the righthand side of (2.8).
Denote
and we have . For given and as parameters, there holds
here we utilize the formula . Then by Lemma 1 we get
by which the series is convergent. Noting that
we deduce that is also convergent by the same arguments. So the derivative is welldefined for .
2.2 Solution to a nonlinear algebraic equation
Consider to solve a group of nonlinear algebraic equations
where is continuous differential on , and is an open bounded domain in . On the local unique solvability of the equation (2.13), we give the following assertion [27].
Lemma 4
For the equation (2.13), assume that there exists a nonsingular matrix , and with a closed sphere such that
where . Then is a homeomorphsim mapping from to .
We now consider to solve a nonlinear algebraic equation in 1D case
where is continuous differential on , and is an open bounded interval in . Based on Lemma 4 there holds
Corollary 1
Suppose that the function is continuous differentiable at , and , and for any as a neighborhood of . Then is a local homeomorphism mapping at , i.e., the equation has a unique solution for given . Moreover, the inverse function is also continuous.
Proof Denote and . By the definition of the derivative, we have
Thus and , there exists such that
Noting , by Lemma 4 follows that the mapping is a local homeomorphism from to , and the equation has a unique solution for given and the inverse function is continuous.
3 The forward problem and the inverse problem
3.1 The forward problem
Consider the homogeneous timefractional diffusion equation in 1D case
where and for . For the equation (3.1), given the homogeneous Dirichelet boundary condition
and the initial distribution
where . A forward problem is composed by the equation (3.1) with the initial boundary value conditions (3.2)(3.3). By using the eigenfunction expansion method, there exists a unique solution (see [39, 46] for instance):
and it can be expressed by the MittagLellfer function given as
where , and , , and is given by (2.7).
3.2 The inverse problem
Although we get the solution to the forward problem given by (3.5), it can not be put into practice if there are unknown parameters in the model. Actually, the fractional order in Eq. (3.1) is an important index characterizing the anomalous diffusion with memory effect in time, but it cannot be measured which leads to inverse fractional order problems.
Suppose that the fractional order in Eq. (3.1) is unknown, and we are to determine it by limited data in time at one space point. Let be fixed, and we have the measured data at time given as
where . The inverse problem is to identify the order using the data (3.6) based on the forward problem (3.1)(3.3).
Denote , and as the unique solution to the forward problem for any given . An interesting problem is:
Can we determine uniquely only using one measurement of at ?
Generally speaking, it is difficult to answer the above question in theory. Nevertheless, we can determine it in some special cases. Denote as the measurement. Then noting the solution’s expression (3.5), we get a nonlinear algebraic equation:
As a result the inverse fractional order problem is transformed to solving of the nonlinear equation (3.7).
4 Unique solvability of the inverse problem
4.1 Local wellposedness
We now prove the inverse problem is locally solvable and has unique solution for by choosing appropriate measured point and the measured time . For that we need the existence of the derivative of the function on , and some limitations for the initial value functions.
Without loss of generality, we set . It is known that the eigenfunction system of the forward problem is for . For , assume that there exists a limited number set such that
which means that there holds
where .
By (3.5) we have
Then the equation (3.7) reduces to
and we get by using (2.8) in Lemma 3
Henceforth, by using Corollary 1 there holds
Theorem 1
For suitable and , there exists such that , and then the equation (4.4) has a unique solution in a neighborhood of , and the solution is continuously dependent upon the measured data locally, i.e., the inverse fractional order problem is of local wellposedness.
Proof By suitably choosing the measured point , we have for , and then there exists one such that .
Then by using Corollary 1, the nonlinear equation (4.4) has a unique solution at the neighborhood of , i.e., the inverse fractional order problem is locally solvable. Moreover, noting the local continuity of the inverse function , the inverse problem is of local stability.
Furthermore, we can get a conditional uniqueness under the nonnegative conditions for the initial values.
4.2 Conditional uniqueness
By Theorem 1 the nonlinear equation (4.4) is locally solvable for where and be fixed. There are no global uniqueness for solving nonlinear equations in general case, but the situation could be changed if special conditions are overposed by which a conditional uniqueness is obtained. For convenience we also set . There holds
Theorem 2
Assume that the the initial value function has the form of (4.2), and there hold for and . Then the fractional order can be uniquely determined by one measurement if and .
Proof Under the given conditions, there are and for . By (4.3) the solution of the forward problem is given as
and we get the nonlinear equation on the order
where is the additional measurement.
By choosing such that for , and noting the condition for , we can deduce that the function is of strict monotonicity on due to the strict monotonicity and positivity of the MittagLelffer function. Therefore the equation (4.7) has only one solution . The proof is completed.
4.3 Theoretical testification
4.3.1 Example 1
In this example we set , and the initial function for , and we choose as the measured point, and as the measured time, i.e., the additional measurement is . Noting and , and , we have by (4.7)
and we need to solve the equation
using the additional data .
Let the exact order be , by which we get the additional data by solving the forward problem. In order to see the unique solvability of the equation, we plot the continuous function on in Figure 1, where and are defined by
It can be seen clearly that the function is strictly monotone on , and the inverse order problem is of uniqueness.
4.3.2 Example 2
We set , and the initial function for in this example. For given , noting that , and , the solution of the forward problem in this example is expressed as follows
For the inverse order problem, let the exact fractional order be . We choose as the measured point, and as the measured time, i.e., the additional measurement is . We have the nonlinear equation combing with the additional data
As done in the above, the function on are plotted in Figure 2.
From Figure 2 it can be seen again that the equation has a unique solution, and the inverse order problem is uniquely solvable by one measurement.
5 Conclusion
An inverse problem of identifying the fractional order in the 1D time fractional diffusion equation by using one measurement is investigated. Based on the expression of the solution to the forward problem, the inverse problem is reduced to a nonlinear algebraic equation on the fractional order, and the unique solvability can be obtained by the monotonicity and positivity of the MittagLellfer function of real variables under the positive conditions for the initial values and the measured point. Theoretical examples are given to illustrate the conditional uniqueness of the inverse problem. It is noted that the derivative of the function on can be computed by (4.8) or (4.12) respectively, and some gradienttype iterative algorithms can be applied to solve the nonlinear equation for which we will give details in another work in the near future.
Acknowledgements
This work is supported by Natural Science Foundation of Shandong Province, China (No. ZR2019MA021), and National Natural Science Foundation of China (nos. 11871313, 11371231).
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