American Institute of Mathematical Sciences

Advanced
December  2018, 7(4): 669-682. doi: 10.3934/eect.2018032

Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method

Zhousheng Ruan 1,2,, , Sen Zhang 2,3, and  Sican Xiong 2,3,

1. 

Fundamental Science on Radioactive Geology and Exploration Technology Laboratory, East China University of Technology, Nanchang, Jiangxi 330013, China
2. 

School of Science, East China University of Technology, Nanchang, Jiangxi 330013, China
3. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Zhousheng Ruan

Received  March 2017 Revised  May 2018 Published  September 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (11661004, 11561003, 11761007), Natural Science Foundation of Jiangxi Province of China (20161BAB201034), Science and Technology Research Project of Education Department of Jiangxi Province (GJJ150568), Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province (20172BCB22019)

In this paper, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. Under some boundedness assumption, the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules, respectively. Based on the superposition principle, we propose a direct inversion algorithm in a parallel manner.

Keywords: Quasi-boundary value method, inverse source problem, convergence rate estimates, time fractional diffusion equation.
Mathematics Subject Classification: Primary: 65M32, 35R30; Secondary: 65M30.
Citation: Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032
References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Academic press, 2003.  Google Scholar

[2]

K. A. Ames and L. E. Payne, Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation, Mathematical Models and Methods in Applied Sciences, 8 (1998), 187-202.  doi: 10.1142/S0218202598000093.  Google Scholar

[3]

J. Cheng, J. Nakagawa, M. Yamamoto, et al., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse problems, 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002.  Google Scholar

[4]

M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, Journal of Mathematical Analysis and Applications, 301 (2005), 419-426.  doi: 10.1016/j.jmaa.2004.08.001.  Google Scholar

[5]

X. L. Feng, L. Eld$\acute{e}$n and C. L. Fu, A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data, Journal of Inverse and Ill-Posed Problems, 18 (2010), 617-645.  doi: 10.1515/JIIP.2010.028.  Google Scholar

[6]

D. N. H$\grave{a}$o, D. N. Van and D. Lesnic, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 055002, 27pp. doi: 10.1088/0266-5611/25/5/055002.  Google Scholar

[7]

D. N. H$\grave{a}$o, D. N. Van and D. Lesnic, Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA Journal of Applied Mathematics, 75 (2010), 291-315.  doi: 10.1093/imamat/hxp026.  Google Scholar

[8]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010, 19pp. doi: 10.1088/0266-5611/28/7/075010.  Google Scholar

[9]

J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788.  doi: 10.1080/00036810903479731.  Google Scholar

[10]

L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8pp. doi: 10.1088/0266-5611/29/7/075013.  Google Scholar

[11]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198, CA: Academic Press Inc, San Diego, 1999.  Google Scholar

[12]

Z. Ruan, Z. Yang and X. Lu, An inverse source problem with sparsity constraint for the timefractional diffusion equation, Advances in Applied Mathematics and Mechanics, 8 (2016), 1-18.  doi: 10.4208/aamm.2014.m722.  Google Scholar

[13]

Z. Ruan and Z. Wang, Identification of a time-dependent source term for a time fractional diffusion problem, Applicable Analysis, 96 (2017), 1638-1655.  doi: 10.1080/00036811.2016.1232400.  Google Scholar

[14]

R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, North-Holland Mathematics Studies, 110 (1985), 421-425.  doi: 10.1016/S0304-0208(08)72739-7.  Google Scholar

[15]

J. G. Wang, Y. B. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Applied Numerical Mathematics, 68 (2013), 39-57.  doi: 10.1016/j.apnum.2013.01.001.  Google Scholar

[16]

Z. Wang, S. Qiu and Z. Ruan, A regularized optimization method for identifying the spacedependent source and the initial value simultaneously in a parabolic equation, Computers & Mathematics with Applications, 67 (2014), 1345-1357.  doi: 10.1016/j.camwa.2014.02.007.  Google Scholar

[17]

L. Wang and J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626.  doi: 10.1016/j.camwa.2012.10.001.  Google Scholar

[18]

J. G. Wang, Y. B. Zhou and T. Wei, A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem, Applied Mathematics Letters, 26 (2013), 741-747.  doi: 10.1016/j.aml.2013.02.006.  Google Scholar

[19]

T. Wei and Z. Q. Zhang, Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Engineering Analysis with Boundary Elements, 37 (2013), 23-31.  doi: 10.1016/j.enganabound.2012.08.003.  Google Scholar

[20]

T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Applied Numerical Mathematics, 78 (2014), 95-111.  doi: 10.1016/j.apnum.2013.12.002.  Google Scholar

[21]

X. T. Xiong, J. X. Wang and M. Li, An optimal method for fractional heat conduction problem backward in time, Applicable Analysis, 91 (2012), 823-840.  doi: 10.1080/00036811.2011.601455.  Google Scholar

[22]

M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems, 28 (2012), 105010, 10pp. doi: 10.1088/0266-5611/28/10/105010.  Google Scholar

[23]

M. Yang and J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Applied Numerical Mathematics, 66 (2013), 45-58.  doi: 10.1016/j.apnum.2012.11.009.  Google Scholar

[24]

X. Ye and C. Xu, Spectral optimization methods for the time fractional diffusion inverse problem, Numer. Math., Theory Methods Appl., 6 (2013), 499-516.   Google Scholar

[25]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011), 035010, 12pp. doi: 10.1088/0266-5611/27/3/035010.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. Fournier, Sobolev Spaces, Academic press, 2003.  Google Scholar

[2]

K. A. Ames and L. E. Payne, Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation, Mathematical Models and Methods in Applied Sciences, 8 (1998), 187-202.  doi: 10.1142/S0218202598000093.  Google Scholar

[3]

J. Cheng, J. Nakagawa, M. Yamamoto, et al., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse problems, 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002.  Google Scholar

[4]

M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, Journal of Mathematical Analysis and Applications, 301 (2005), 419-426.  doi: 10.1016/j.jmaa.2004.08.001.  Google Scholar

[5]

X. L. Feng, L. Eld$\acute{e}$n and C. L. Fu, A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data, Journal of Inverse and Ill-Posed Problems, 18 (2010), 617-645.  doi: 10.1515/JIIP.2010.028.  Google Scholar

[6]

D. N. H$\grave{a}$o, D. N. Van and D. Lesnic, A non-local boundary value problem method for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 055002, 27pp. doi: 10.1088/0266-5611/25/5/055002.  Google Scholar

[7]

D. N. H$\grave{a}$o, D. N. Van and D. Lesnic, Regularization of parabolic equations backward in time by a non-local boundary value problem method, IMA Journal of Applied Mathematics, 75 (2010), 291-315.  doi: 10.1093/imamat/hxp026.  Google Scholar

[8]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28 (2012), 075010, 19pp. doi: 10.1088/0266-5611/28/7/075010.  Google Scholar

[9]

J. J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89 (2010), 1769-1788.  doi: 10.1080/00036810903479731.  Google Scholar

[10]

L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8pp. doi: 10.1088/0266-5611/29/7/075013.  Google Scholar

[11]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198, CA: Academic Press Inc, San Diego, 1999.  Google Scholar

[12]

Z. Ruan, Z. Yang and X. Lu, An inverse source problem with sparsity constraint for the timefractional diffusion equation, Advances in Applied Mathematics and Mechanics, 8 (2016), 1-18.  doi: 10.4208/aamm.2014.m722.  Google Scholar

[13]

Z. Ruan and Z. Wang, Identification of a time-dependent source term for a time fractional diffusion problem, Applicable Analysis, 96 (2017), 1638-1655.  doi: 10.1080/00036811.2016.1232400.  Google Scholar

[14]

R. E. Showalter, Cauchy problem for hyper-parabolic partial differential equations, North-Holland Mathematics Studies, 110 (1985), 421-425.  doi: 10.1016/S0304-0208(08)72739-7.  Google Scholar

[15]

J. G. Wang, Y. B. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Applied Numerical Mathematics, 68 (2013), 39-57.  doi: 10.1016/j.apnum.2013.01.001.  Google Scholar

[16]

Z. Wang, S. Qiu and Z. Ruan, A regularized optimization method for identifying the spacedependent source and the initial value simultaneously in a parabolic equation, Computers & Mathematics with Applications, 67 (2014), 1345-1357.  doi: 10.1016/j.camwa.2014.02.007.  Google Scholar

[17]

L. Wang and J. Liu, Data regularization for a backward time-fractional diffusion problem, Computers & Mathematics with Applications, 64 (2012), 3613-3626.  doi: 10.1016/j.camwa.2012.10.001.  Google Scholar

[18]

J. G. Wang, Y. B. Zhou and T. Wei, A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward time-fractional diffusion problem, Applied Mathematics Letters, 26 (2013), 741-747.  doi: 10.1016/j.aml.2013.02.006.  Google Scholar

[19]

T. Wei and Z. Q. Zhang, Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Engineering Analysis with Boundary Elements, 37 (2013), 23-31.  doi: 10.1016/j.enganabound.2012.08.003.  Google Scholar

[20]

T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Applied Numerical Mathematics, 78 (2014), 95-111.  doi: 10.1016/j.apnum.2013.12.002.  Google Scholar

[21]

X. T. Xiong, J. X. Wang and M. Li, An optimal method for fractional heat conduction problem backward in time, Applicable Analysis, 91 (2012), 823-840.  doi: 10.1080/00036811.2011.601455.  Google Scholar

[22]

M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems, 28 (2012), 105010, 10pp. doi: 10.1088/0266-5611/28/10/105010.  Google Scholar

[23]

M. Yang and J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Applied Numerical Mathematics, 66 (2013), 45-58.  doi: 10.1016/j.apnum.2012.11.009.  Google Scholar

[24]

X. Ye and C. Xu, Spectral optimization methods for the time fractional diffusion inverse problem, Numer. Math., Theory Methods Appl., 6 (2013), 499-516.   Google Scholar

[25]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems, 27 (2011), 035010, 12pp. doi: 10.1088/0266-5611/27/3/035010.  Google Scholar

Figure 1.  Error curves for example 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Error surfaces for example 2
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Inversional results for example 1 with different relative errors
$\delta$ 0.05% 0.1% 0.2% 0.4% 0.8% 1.6%
$T_1$=0.1 $e(f, \delta)$ 2.1% 2.9% 4.0% 5.5% 7.9% 10.1%
$C_r$ 0.34 0.33 0.31 0.38 0.24
$T_1$=0.2 $e(f, \delta)$ 2.1% 3.1% 4.1% 5.9% 8.3% 11.0%
$C_r$ 0.38 0.28 0.35 0.33 0.28
$T_1$=0.4 $e(f, \delta)$ 2.2% 3.2% 4.2% 6.2% 8.5 % 11.5%
$C_r$ 0.39 0.26 0.37 0.31 0.30
$T_1$=0.8 $e(f, \delta)$ 2.2% 3.3 % 4.3% 6.3% 8.6 % 11.8%
$C_r$ 0.40 0.25 0.38 0.30 0.31
$T_1$=1 $e(f, \delta)$ 2.2% 3.3 % 4.3% 6.4% 8.6 % 11.8%
$C_r$ 0.40 0.25 0.38 0.29 0.31
Table Options

Download as excel

$\delta$ 0.05% 0.1% 0.2% 0.4% 0.8% 1.6%
$T_1$=0.1 $e(f, \delta)$ 2.1% 2.9% 4.0% 5.5% 7.9% 10.1%
$C_r$ 0.34 0.33 0.31 0.38 0.24
$T_1$=0.2 $e(f, \delta)$ 2.1% 3.1% 4.1% 5.9% 8.3% 11.0%
$C_r$ 0.38 0.28 0.35 0.33 0.28
$T_1$=0.4 $e(f, \delta)$ 2.2% 3.2% 4.2% 6.2% 8.5 % 11.5%
$C_r$ 0.39 0.26 0.37 0.31 0.30
$T_1$=0.8 $e(f, \delta)$ 2.2% 3.3 % 4.3% 6.3% 8.6 % 11.8%
$C_r$ 0.40 0.25 0.38 0.30 0.31
$T_1$=1 $e(f, \delta)$ 2.2% 3.3 % 4.3% 6.4% 8.6 % 11.8%
$C_r$ 0.40 0.25 0.38 0.29 0.31
Table 2.  Inversional results for example 2 with different relative errors
$\delta$ 0.05% 0.1% 0.2% 0.4% 0.8% 1.6%
$T_1$=0.1 $e(f, \delta)$ 1.9% 2.8% 4.7% 6.9% 10.2% 14.7%
$C_r$ 0.41 0.50 0.37 0.39 0.36
$T_1$=0.2 $e(f, \delta)$ 1.9% 2.9% 4.7% 6.9% 10.3% 14.8%
$C_r$ 0.41 0.49 0.37 0.39 0.35
$T_1$=0.4 $e(f, \delta)$ 1.9% 2.9% 4.8% 7.0% 10.4 % 14.9%
$C_r$ 0.40 0.50 0.37 0.39 0.35
$T_1$=0.8 $e(f, \delta)$ 1.9% 2.9 % 4.8% 7.0% 10.5 % 14.9%
$C_r$ 0.40 0.50 0.38 0.40 0.34
$T_1$=1 $e(f, \delta)$ 1.9% 2.9 % 4.8% 7.0% 10.5 % 14.9%
$C_r$ 0.40 0.50 0.38 0.40 0.34
Table Options

Download as excel

$\delta$ 0.05% 0.1% 0.2% 0.4% 0.8% 1.6%
$T_1$=0.1 $e(f, \delta)$ 1.9% 2.8% 4.7% 6.9% 10.2% 14.7%
$C_r$ 0.41 0.50 0.37 0.39 0.36
$T_1$=0.2 $e(f, \delta)$ 1.9% 2.9% 4.7% 6.9% 10.3% 14.8%
$C_r$ 0.41 0.49 0.37 0.39 0.35
$T_1$=0.4 $e(f, \delta)$ 1.9% 2.9% 4.8% 7.0% 10.4 % 14.9%
$C_r$ 0.40 0.50 0.37 0.39 0.35
$T_1$=0.8 $e(f, \delta)$ 1.9% 2.9 % 4.8% 7.0% 10.5 % 14.9%
$C_r$ 0.40 0.50 0.38 0.40 0.34
$T_1$=1 $e(f, \delta)$ 1.9% 2.9 % 4.8% 7.0% 10.5 % 14.9%
$C_r$ 0.40 0.50 0.38 0.40 0.34
[1]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[2]

Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
[3]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019045
[4]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007
[5]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[6]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[7]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927
[8]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007
[9]

Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033
[10]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
[11]

Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767
[12]

Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas and Michael Taylor. Metric tensor estimates, geometric convergence, and inverse boundary problems. Electronic Research Announcements, 2003, 9: 69-79.

[13]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[14]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[15]

Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029
[16]

Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107
[17]

Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $ x $-dependent coefficients. Evolution Equations & Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001
[18]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007
[19]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[20]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (114)
  • HTML views (354)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences