Abstract
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 illposedness 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.
1 Introduction
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 [4]. It has the following forms:
where
with
and
where
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 illposed. Regularization technique has to be introduced to obtain stable numerical solution. A few progress has been developed for solving the inverse source problems of spacefractional diffusion equation [4,14,15, 16,17]. In [16], 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 [18] and the Cauchy problem of the Laplace equation [19]. 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
It is well known that the norm of Sobolev space
The Fourier transform of
From [4],
where
We can deduce the following equation in Fourier space by using the the Fourier transform to problem (1):
where
We have a solution to problem (4) in the following form:
We define
and then it is easy to see that
2.2 The modified Tikhonov regularization method
Let
The Hermite expansion of a function
where
Let
where
Suppose that
where
If
is chosen as the approximation of
Lemma 1
If we let
The function
and
3 Error estimate of the method
In this section, we deduce the error estimate of the new method. The following auxiliary results are needed.
Lemma 2
[4] For
where
Lemma 3
[16] For any
Suppose that the Fourier Hermite coefficients vector of
then we let
where
Lemma 4
If
where
Proof
According to Parseval’s formula and Lemma 2, we obtain
Lemma 5
Let
Then there exists a constant
Proof
Let
According to Parseval’s formula, we have the following results:
and
This completes the proof.□
Theorem 6
If
and then we have
Proof
Due to (2), (16), and (19), we can obtain the following result by using the triangle inequality:
If we define
Hence, by using the triangle inequality, (6), (16), (22), and Lemma 1,
Let
Suppose that
then
holds with constants
So we can obtain
Due to (2) and (19), we can obtain the following result by using triangle inequality:
The assertion can be obtained by using (24), (25), and Lemma 3.□
4 Numerical experiments
To verify the effectiveness of the proposed method, we present some numerical tests in this section. For simplicity, let
We perform the numerical tests in a finite interval
where
The analytical solution of equation (1) is usually difficult to obtain, so we have to use the numerical method to obtain the datum
Moreover, we would like to compare the relative errors obtained by the method in this article (M1) with that in [16] (M2). All the results are obtained by using Matlab2017b in the case of
Example 1
We take the function
and the numerical text is implemented in the interval
The comparisons of the exact function and its approximations for various






M1  M2  M1  M2  M1  M2  

0.0729  0.0684  0.1588  0.1934  0.2102  0.4258 

0.0093  0.0087  0.0376  0.0581  0.0554  0.1584 

0.0018  0.0020  0.0057  0.0095  0.0117  0.0222 
Example 2
Consider a nonsmooth function:
The errors are given in Table 2, and the comparisons of the exact function and its approximations for various






M1  M2  M1  M2  M1  M2  

0.1046  0.1065  0.1732  0.2256  0.2509  0.4522 

0.0128  0.0135  0.0548  0.0846  0.0782  0.2126 

0.0082  0.0094  0.0157  0.0213  0.0468  0.0952 
5 Conclusion
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 illposed problems.
Acknowledgements
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.
References
[1] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2001), no. 4, 230–237. Search in Google Scholar
[2] R. Metzler and J. Klafter, The random walkas guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1–77. 10.1016/S03701573(00)000703Search in Google Scholar
[3] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, R161. 10.1088/03054470/37/31/R01Search in Google Scholar
[4] W. Y. Tian, C. Li, W. H. Deng, and Y. J. Wu, Regularization methods for unknown source in space fractional diffusion equation, Math. Comput. Simulation 85 (2012), 45–56. 10.1016/j.matcom.2012.08.011Search in Google Scholar
[5] J. R. Cannon and P. Duchateau, Structural identification of an unknown source term in a heat equation, Inverse Problems 14 (1999), no. 3, 535–551. 10.1088/02665611/14/3/010Search in Google Scholar
[6] S. Andrieux and A. B. Abda, Identification of planar cracks by complete overdetermined data: inversion formulae, Inverse Problems 12 (1999), no. 5, 553. 10.1088/02665611/12/5/002Search in Google Scholar
[7] J. I. Frankel, Residualminimization leastsquares method for inverse heat conduction, Comput. Math. Appl. 32 (1996), no. 4, 117–130. 10.1016/08981221(96)001307Search in Google Scholar
[8] A. J. Devaney and E. A. Marengo, Inverse source problem in nonhomogeneous background media, SIAM J. Appl. Math. 67 (2007), no. 5, 1353–1378. 10.1137/060658618Search in Google Scholar
[9] R. A. Albanese and P. B. Monk, The inverse source problem for Maxwell’s equations, Inverse Problems 22 (2006), no. 3, 1023–1035. 10.21236/ADA459256Search in Google Scholar
[10] H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo and photoacoustic tomography, Inverse Problems 31 (2015), no. 10, 105005. 10.1088/02665611/31/10/105005Search in Google Scholar
[11] M. Eller and N. P. Valdivia, Acoustic source identification using multiple frequency information, Inverse Problems 28 (2009), no. 11, 1183–1223. 10.1088/02665611/25/11/115005Search in Google Scholar
[12] G. Wang, F. M. Ma, Y. K. Guo, and J. Z. Li, Solving the multifrequency electromagnetic inverse source problem by the Fourier method, J. Differential Equations 265 (2018), 417–443. 10.1016/j.jde.2018.02.036Search in Google Scholar
[13] N. Magnoli and G. A. Viano, The source identification problem in electromagnetic theory, J. Math. Phys. 38 (1997), no. 5, 2366–2388. 10.1063/1.531978Search in Google Scholar
[14] H. Wei, W. Chen, H. Sun, and X. Li, A coupled method for inverse source problem of spatial fractional anomalous diffusion equations, Inverse Probl. Sci. Eng. 18 (2010), no. 7, 945–956. 10.1080/17415977.2010.492515Search in Google Scholar
[15] F. Yang, C. L. Fu, and X. X. Li, Identifying an unknown source in spacefractional diffusion equation, Acta Math. Sci. 34B (2014), no. 4, 1012–1024. 10.1016/S02529602(14)600655Search in Google Scholar
[16] Z. Y. Zhao, O. Xie, L. You, and Z. H. Meng, A truncation method based on Hermite expansion for unknown source in space fractional diffusion equation, Math. Model. Anal. 19 (2014), no. 3, 430–442. 10.3846/13926292.2014.929057Search in Google Scholar
[17] D. D. Trong, D. Hai, and N. D. Minh, Optimal regularization for an unknown source of spacefractional diffusion equation, Appl. Math. Comput. 349 (2019), 184–206. 10.1016/j.amc.2018.12.030Search in Google Scholar
[18] Z. Y. Zhao, A Hermite extension method for numerical differentiation, Appl. Numer. Math. 159 (2021), 46–60. 10.1016/j.apnum.2020.08.016Search in Google Scholar
[19] Z. Y. Zhao, L. You, and Z. H. Meng, A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation, Open Math. 18 (2020), no. 1, 1685–1697. 10.1515/math20200111Search in Google Scholar
[20] I. Podlubny, Fractional Differential Equations, Elsevier, Amsterdam, 1998. Search in Google Scholar
[21] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, 1993. 10.1515/9780691213927Search in Google Scholar
[22] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Springer Science & Business Media, Berlin, 2000. 10.1007/9789400917408_3Search in Google Scholar
[23] M. T. Nair, S. V. Pereverzev, and U. Tautenhahn, Regularization in Hilbert scales under general smoothing conditions, Inverse Problems 21 (2005), no. 6, 1851. 10.1088/02665611/21/6/003Search in Google Scholar
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