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Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results
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 |
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.
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. |
show all references
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. |


0.05% | 0.1% | 0.2% | 0.4% | 0.8% | 1.6% | ||
2.1% | 2.9% | 4.0% | 5.5% | 7.9% | 10.1% | ||
0.34 | 0.33 | 0.31 | 0.38 | 0.24 | |||
2.1% | 3.1% | 4.1% | 5.9% | 8.3% | 11.0% | ||
0.38 | 0.28 | 0.35 | 0.33 | 0.28 | |||
2.2% | 3.2% | 4.2% | 6.2% | 8.5 % | 11.5% | ||
0.39 | 0.26 | 0.37 | 0.31 | 0.30 | |||
2.2% | 3.3 % | 4.3% | 6.3% | 8.6 % | 11.8% | ||
0.40 | 0.25 | 0.38 | 0.30 | 0.31 | |||
2.2% | 3.3 % | 4.3% | 6.4% | 8.6 % | 11.8% | ||
0.40 | 0.25 | 0.38 | 0.29 | 0.31 |
0.05% | 0.1% | 0.2% | 0.4% | 0.8% | 1.6% | ||
2.1% | 2.9% | 4.0% | 5.5% | 7.9% | 10.1% | ||
0.34 | 0.33 | 0.31 | 0.38 | 0.24 | |||
2.1% | 3.1% | 4.1% | 5.9% | 8.3% | 11.0% | ||
0.38 | 0.28 | 0.35 | 0.33 | 0.28 | |||
2.2% | 3.2% | 4.2% | 6.2% | 8.5 % | 11.5% | ||
0.39 | 0.26 | 0.37 | 0.31 | 0.30 | |||
2.2% | 3.3 % | 4.3% | 6.3% | 8.6 % | 11.8% | ||
0.40 | 0.25 | 0.38 | 0.30 | 0.31 | |||
2.2% | 3.3 % | 4.3% | 6.4% | 8.6 % | 11.8% | ||
0.40 | 0.25 | 0.38 | 0.29 | 0.31 |
0.05% | 0.1% | 0.2% | 0.4% | 0.8% | 1.6% | ||
1.9% | 2.8% | 4.7% | 6.9% | 10.2% | 14.7% | ||
0.41 | 0.50 | 0.37 | 0.39 | 0.36 | |||
1.9% | 2.9% | 4.7% | 6.9% | 10.3% | 14.8% | ||
0.41 | 0.49 | 0.37 | 0.39 | 0.35 | |||
1.9% | 2.9% | 4.8% | 7.0% | 10.4 % | 14.9% | ||
0.40 | 0.50 | 0.37 | 0.39 | 0.35 | |||
1.9% | 2.9 % | 4.8% | 7.0% | 10.5 % | 14.9% | ||
0.40 | 0.50 | 0.38 | 0.40 | 0.34 | |||
1.9% | 2.9 % | 4.8% | 7.0% | 10.5 % | 14.9% | ||
0.40 | 0.50 | 0.38 | 0.40 | 0.34 |
0.05% | 0.1% | 0.2% | 0.4% | 0.8% | 1.6% | ||
1.9% | 2.8% | 4.7% | 6.9% | 10.2% | 14.7% | ||
0.41 | 0.50 | 0.37 | 0.39 | 0.36 | |||
1.9% | 2.9% | 4.7% | 6.9% | 10.3% | 14.8% | ||
0.41 | 0.49 | 0.37 | 0.39 | 0.35 | |||
1.9% | 2.9% | 4.8% | 7.0% | 10.4 % | 14.9% | ||
0.40 | 0.50 | 0.37 | 0.39 | 0.35 | |||
1.9% | 2.9 % | 4.8% | 7.0% | 10.5 % | 14.9% | ||
0.40 | 0.50 | 0.38 | 0.40 | 0.34 | |||
1.9% | 2.9 % | 4.8% | 7.0% | 10.5 % | 14.9% | ||
0.40 | 0.50 | 0.38 | 0.40 | 0.34 |
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