# American Institute of Mathematical Sciences

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

 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.

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] R. A. Adams and J. Fournier, Sobolev Spaces, Academic press, 2003. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
Error curves for example 1
Error surfaces for example 2
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
 $\delta$ 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
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
 $\delta$ 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
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