In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the method.
Due to its wide application in many fields, including physical, and mechanical engineering, signal processing and systems identification, control theory, and finance, fractional differential equations have received extensive attention during the past decades. One of the most important applications of fractional differential equations is that it can characterize the abnormal diffusion effectively. For a review, the reader can refer to [1,2,3] and the references therein.
In this article, we consider a space fractional diffusion equation that can describe the probability distribution of particles with superdiffusion . It has the following forms:
with , and . For . We define the right Riemann–Liouville fractional derivatives and the left Riemann–Liouville fractional derivatives as follows:
where is the smallest integer no less than . Our goal is to identify the source function in (1) from the additional observed data . Generally, the data is measured, and we only have the noisy data that satisfies
This problem is referred as an inverse source problem.
The inverse problems occur in many fields: for example, crack identification, heat conduction problems, pollutant detection, target tracking, and antenna synthesis [5,6,7, 8,9,10, 11,12,13]. The main difficulty of the inverse source problem is usually ill-posed. Regularization technique has to be introduced to obtain stable numerical solution. A few progress has been developed for solving the inverse source problems of space-fractional diffusion equation [4,14,15, 16,17]. In , a truncated Hermite expansion method has been proposed to deal with problem (1). The method is effective, but the source condition for deriving convergence result is not natural, and the convergence rate of the method is not ordered optimal. To overcome the aforementioned shortcomings, we proposed a modified Tikhonov regularization based on Hermite expansion for problem (1). This method has been successfully applied to solving the numerical differentiation problem  and the Cauchy problem of the Laplace equation . For the new method, we can obtain the convergence results under a weaker condition, and the convergence rates is the order optimal when the regularization parameter is chosen by the discrepancy principle.
This article has the following structure. In Section 2, we describe the process of this method. Section 3 is about the error estimate of the method. To verify the effectiveness of the method, we also give some numerical experiments in Section 4.
2 Basic description of the method
2.1 Problem of the solution
Let and are the usual Lebesgue and Sobolev spaces on , and denote the norms in . We define the Fourier transform of the function as follows:
It is well known that the norm of Sobolev space can be defined as follows:
The Fourier transform of can be given as follows :
where represents the signum function.
We can deduce the following equation in Fourier space by using the the Fourier transform to problem (1):
We have a solution to problem (4) in the following form:
and then it is easy to see that
is unbounded as tends to infinity, so the problem is difficult to solve. Small errors can have a huge impact on the results. Therefore, special regularization technique is required to deal with it.
2.2 The modified Tikhonov regularization method
Let be the normalized Hermite function of degree . According to , the Hermite functions have the following orthogonality relations:
The Hermite expansion of a function can be given as follows:
Let we define operators
where is the characteristic function of the interval .
where is a constant. Now we devote to develop a method to obtain a stable approximation of from the noisy data . Let and . We propose a modified Tikhonov functional of the following form:
If is the minimizer of above functional, then
is chosen as the approximation of . It can be deduced that can be obtained by solving the following equation :
If we let , then
The function has the following properties :
3 Error estimate of the method
In this section, we deduce the error estimate of the new method. The following auxiliary results are needed.
 For , we have
where and are two constants.
 For any , if , then
Suppose that the Fourier Hermite coefficients vector of is , i.e.,
then we let
where is a positive integer that has to be chosen properly. (It should be noted that is only used in theoretical analysis.)
If , then we have
According to Parseval’s formula and Lemma 2, we obtain
Let ( ) is some fixed constants, if the vector sequence satisfies
Then there exists a constant such that
Let , then by using the triangle inequality, we have
According to Parseval’s formula, we have the following results:
This completes the proof.□
and then we have
If we define , then we have
Suppose that satisfies
holds with constants . Hence, according to Lemma 5, there exists a constant ,
So we can obtain
4 Numerical experiments
To verify the effectiveness of the proposed method, we present some numerical tests in this section. For simplicity, let , , , and = 1. We also test the effect of the method when the parameters are different, and the result is similar.
We perform the numerical tests in a finite interval , and approaches zero as . Let the knots with . The datum represents values of on the grid. Then the perturbation data is obtained by adding random uniformly distributed perturbation to , i.e.,
where is the noise level and the noise is measured by
The analytical solution of equation (1) is usually difficult to obtain, so we have to use the numerical method to obtain the datum for a given . This step is similar to the method presented in . The following relative error of norm is used to measure the accuracy of the numerical approximation:
We take the function as follows:
and the numerical text is implemented in the interval .
The comparisons of the exact function and its approximations for various with are given in Figure 1. The relative errors of M1 and M2 for various and are presented in Table 1. It can be seen that when alpha is close to 1, the results of the two methods are close. With the increase of , method 1 performs better than method 2, and the numerical results are more stable and the relative errors are smaller. Figures 2 and 3 exhibit the variation of with the changes of and , respectively. It can be seen that the method is stable for various and .
Consider a nonsmooth function:
The errors are given in Table 2, and the comparisons of the exact function and its approximations for various with are shown in Figure 4. It can be seen that the method is still stable, and the results of M1 are also better than M2 in this case.
On the basis of the Hermite extension method, we propose a modified Tikhonov regularization to solve an unknown source problem in the space fractional diffusion equation. In addition, the framework of this approach can be used to deal with other ill-posed problems.
The authors thank the referees for valuable comments and suggestions, which improved the presentation of this manuscript.
Funding information: The project is supported by the project of enhancing school with innovation of Guangdong ocean university (Q18306).
Conflict of interest: The authors state no conflict of interest.
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