- Research
- Open Access
The quasi-boundary value regularization method for identifying the initial value with discrete random noise
- Fan Yang^{1}Email author,
- Yan Zhang^{1},
- Xiao-Xiao Li^{1} and
- Can-Yun Huang^{1}
- Received: 14 March 2018
- Accepted: 3 July 2018
- Published: 7 July 2018
Abstract
In this paper, we study an inverse initial value problem for the fractional diffusion equation with discrete noise. This problem is ill-posed in the sense of Hadamard. We apply the trigonometric method in a nonparametric regression associated with the quasi-boundary value regularization method to deal with this ill-posed problem. The corresponding convergence estimate for this method is obtained. The numerical results show that this regularization method is flexible and stable.
Keywords
- Time-fractional diffusion equation
- Quasi-boundary value regularization method
- Nonparametric regression
- Discrete random data
MSC
- 35R25
- 47A52
- 35R30
1 Introduction
In recent years, fractional differential equations have attracted worldwide attention due to their wide applications in different research areas and engineering, such as physical [1, 2], chemical [3], biology [4], signal processing [5], mechanical engineering [6] and systems identification [7], electrical and fractional dynamics [8–10]. However, for some practical situations, the part of the diffusion coefficient, or initial data, or boundary data, or source term may not be known, we need to find them using some additional measurement data, which will lead to the inverse problem of the fractional diffusion equation, such as [11–13]. Recently, many researchers have presented results of the initial value problem and boundary value problem on fractional differential equations, such as [14–16]. In [17], the authors used the monotone iterative method to consider the existence and uniqueness of solution of the initial value problem for a fractional differential equation. In [18], the authors used quasi-reversible method to consider initial value problem for a time-fractional diffusion equation. In [19], the authors used a modified quasi-boundary value method to determine the initial data from a noisy final data in a time-fractional diffusion equation. Above these references on identifying the initial value of fractional diffusion equations, the measurable data is selected as a continuous function. However, in practice, the measure data is always discrete. The discrete random data is closer to practice. To the best of our knowledge, there were few papers for identifying the initial value of fractional diffusion equations with the discrete random data. In [20], the authors once used the truncation regularization method to identify the unknown source for a time-fractional equation with the discrete random noise, but we consider the inverse initial value problem with this special type of noise in the data.
In this paper, we extend this discrete random noise to identify the initial value problem by the quasi-boundary value regularization method. In [22], the quasi-boundary value method was first called non-local boundary value problem method and was used to solve the backward heat conduction problem. Wei and Wang in [19] used the quasi-boundary value regularization method to deal with the backward problem. Now, this method is also studied for solving various types of inverse problems, such as parabolic equations [22–24], hyper-parabolic equations [25], and elliptic equations [26].
The general structure of this paper is as follows: we first present some preliminary results in Sect. 2. In Sect. 3 we develop the trigonometric method in nonparametric regression associated with quasi-boundary value regularization method to construct the regularized solution. Section 4 contains the convergence estimate under an a priori assumption for the exact solution. Some numerical results are presented in Sect. 5. Section 6 is a brief conclusion.
2 Preliminaries
In this section, we introduce some useful definitions and preliminary results.
Definition 2.1
([27])
Lemma 2.1
([27])
Lemma 2.2
([28])
Lemma 2.3
([29])
Lemma 2.4
([30], page 144)
Lemma 2.5
Proof
3 Regularized solutions for backward problem for time-fractional diffusion equation
4 Estimators and convergence results
Lemma 4.1
Proof
Lemma 4.2
The proof is very easy and we omit it here.
The main result of this section is the following.
Theorem 4.1
Proof
Remark 4.1
Remark 4.2
Remark 4.3
5 Numerical results
Example
Numerical results of the example for different \(\sigma^{2}\) with \(\alpha=0.2\)
\(\sigma^{2}=10^{-4}\) | \(\sigma^{2}=10^{-5}\) | |
---|---|---|
\(e_{r}\) | 0.0971 | 0.0116 |
μ | 5.6555e − 006 | 5.6485e − 007 |
Numerical results of the example for different \(\sigma^{2}\) with \(\alpha=0.8\)
\(\sigma^{2}=10^{-4}\) | \(\sigma^{2}=10^{-5}\) | |
---|---|---|
\(e_{r}\) | 0.2485 | 0.0249 |
μ | 1.4683e − 008 | 1.4697e − 009 |
6 Conclusion
In this paper, we solve the inverse initial value problem for a time-fractional diffusion equation. The trigonometric method in nonparametric regression associated with the quasi-boundary value regularization method is applied to solve the ill-posed problem. Specially, the problem is dealt with the discrete random noise. The convergence estimate is presented under an a priori regularization parameter choice rule. In numerical experiments, the computational cost is within 10 seconds and the convergence results is proved, so this work is good. In future work, we will continue to research the other inverse problems of this special type of noise in the data, such as identifying the source of the space-fractional diffusion equation.
Declarations
Acknowledgements
The authors would like to thanks the editor and the referees for their valuable comments and suggestions that improve the quality of our paper.
Availability of data and materials
Not applicable.
Funding
The work is supported by the National Natural Science Foundation of China (11561045,11501272) and the Doctor Fund of Lan Zhou University of Technology.
Authors’ contributions
The main idea of this paper was proposed by FY and YZ prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Phys. A, Stat. Mech. Appl. 278(1), 107–125 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Autuori, G., Cluni, F., Gusella, V., Pucci, P.: Mathematical models for nonlocal elastic composite materials. Adv. Nonlinear Anal. 6(4), 355–382 (2017) MathSciNetMATHGoogle Scholar
- Ghergu, M., Radulescu, V.D.: Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics. Springer Monographs in Mathematics, Springer, Heidelberg (2012) MATHGoogle Scholar
- Yuste, S.B., Lindenberg, K.: Subdiffusion-limited reactions. Chem. Phys. 284(1), 169–180 (2002) View ArticleGoogle Scholar
- Kumar, S., Kumar, D., Singh, J.: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal. 5(4), 383–394 (2016) MathSciNetMATHGoogle Scholar
- Magin, R., Feng, X., Baleanu, D.: Solving the fractional order Bloch equation. Concepts Magn. Reson., Part A, Bridg. Educ. Res. 34(1), 16–23 (2009) View ArticleGoogle Scholar
- Duncan, T.E., Pasik-Duncan, B.: A direct approach to linear-quadratic stochastic control. Opusc. Math. 37(6), 821–827 (2017) MathSciNetView ArticleGoogle Scholar
- Jin, B.T., Rundell, W.: A tutorial on inverse problems for anomalous diffusion processes. Inverse Probl. 31(3), 035003 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Marin, M., Dumitru, B.: On vibrations in thermoelasticity without energy dissipation for micropolar bodies. Bound. Value Probl. 2016, 111 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Radulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015) View ArticleMATHGoogle Scholar
- Wang, J.G., Zhou, Y.B., Wei, T.: Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation. Appl. Numer. Math. 68, 39–57 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Yang, F., Fu, C.L.: The quasi-reversibility regularization method for identifying the unknown source for time-fractional diffusion equation. Appl. Math. Model. 39, 1500–1512 (2015) MathSciNetView ArticleGoogle Scholar
- Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Bachar, I., Mâagli, H., Radulescu, V.D.: Fractional Navier boundary value problems. Bound. Value Probl. 79, 14 (2016) MathSciNetMATHGoogle Scholar
- Bachar, I., Mâagli, H., Radulescu, V.D.: Positive solutions for superlinear Riemann–Liouville fractional boundary-value problems. Electron. J. Differ. Equ. 240, 16 (2017) MathSciNetMATHGoogle Scholar
- Denton, Z., Ramírez, J.D.: Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems. Opusc. Math. 37(5), 705–724 (2017) MathSciNetView ArticleGoogle Scholar
- Wei, Z.L., Li, Q.D., Che, J.L.: Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, 260–272 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Yang, M., Liu, J.J.: Solving a final value fractional diffusion problem by boundary condition regularization. Appl. Numer. Math. 66, 45–58 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Wei, Y., Wang, J.G.: A modified quasi-boundary value method for the backward time-fractional diffusion problem. Math. Model. Numer. Anal. 48, 603–621 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Tuan, N.H., Nane, E.R.: Inverse source problem for time fractional diffusion with discrete random noise. Stat. Probab. Lett. 120, 126–134 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999) MATHGoogle Scholar
- Hao, D.N., Duc, N.V., Lesnic, D.: Regularization of parabolic equations backward in time by a non-lacal boundary value problem method. Appl. Math. (Irvine) 75, 291–315 (2010) MATHGoogle Scholar
- Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301, 419–426 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Hao, D.N., Duc, N.V., Sahli, H.: A non-local boundary value problem method for parabolic equations backward in time. J. Math. Anal. Appl. 345, 805–815 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Showalter, R.E.: Cauchy problem for hyper-partial differential equations. North-Holl. Math. Stud. 110, 421–425 (1985) View ArticleMATHGoogle Scholar
- Feng, X.L., Elden, L., Fu, C.L.: A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data. J. Inverse Ill-Posed Probl. 18, 617–645 (2010) MathSciNetMATHGoogle Scholar
- Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999) MATHGoogle Scholar
- Liu, J.J., Yamamoto, M.: A backward problem for the time-fractional diffusion equation. Appl. Anal. 89(11), 1769–1788 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Wang, J.G., Zhou, Y.B., Wei, T.: A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem. Appl. Math. Lett. 26(7), 741–747 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Eubank, R.L.: Nonparametric Regression and Spline Smoothing, 2nd edn. Statistics: Textbooks and Monographs, vol. 157. Dekker, New York (1999) MATHGoogle Scholar