On an inverse problem for fractional evolution equation

Nguyen Huy Tuan; Mokhtar Kirane; Long Dinh Le; Van Thinh Nguyen

doi:10.3934/eect.2017007

Article Contents

2017, Volume 6, Issue 1: 111-134. Doi: 10.3934/eect.2017007

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On an inverse problem for fractional evolution equation

1.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam
2.
Laboratoire de Mathématiques Pôle Sciences et Technologie, Universié de La Rochelle, Aénue M. Crépeau, 17042 La Rochelle Cedex, France
3.
Institute of Computational Science and Technology, Ho Chi Minh City, Viet Nam
4.
Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea

* Corresponding author:nguyenhuytuan@tdt.edu.vn.
* Corresponding author:nguyenhuytuan@tdt.edu.vn.

Received: January 31, 2016

Revised: August 31, 2016

Published: March 2018

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Abstract

In this paper, we investigate a backward problem for a fractional abstract evolution equation for which we wants to extract the initial distribution from the observation data provided along the final time $t = T.$ This problem is well-known to be ill-posed due to the rapid decay of the forward process. We consider a final value problem for fractional evolution process with respect to time. For this ill-posed problem, we construct two regularized solutions using quasi-reversibility method and quasi-boundary value method. The well-posedness of the regularized solutions as well as the convergence property is analyzed. The advantage of the proposed methods is that the regularized solution is given analytically and therefore is easy to be implemented. A numerical example is presented to show the validity of the proposed methods.

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Mathematics Subject Classification: 35K05, 35K99, 47J06, 47H10x.

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Figure 1. Reconstruction results at t = 0 from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

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Figure 2. Reconstruction results at t = 0 from noisy measurement data: 2D drawing using QBV Method

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Figure 3. Reconstruction results at $t = 0.05$ from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

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Figure 4. Reconstruction results at $t = 0.25$ from noisy measurement data at $T = 2$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using QBV method

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Figure 5. Reconstruction results at t = 0 from noisy measurement data at $t=0$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using Quasi Reversibility method

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Figure 6. Reconstruction results at t = 0 from noisy measurement data at $t=0$ with $ \in =10^{-1}, \in =10^{-2}, \in = 10^{-3}$ using Quasi Reversibility method

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Figure 7. The exact solution in Example 2 at t = 1.

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Figure 8. Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $ \in =10^{-1}, \in =10^{-2}$ using Quasi Reversibility method

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Figure 9. Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $ \in = 10^{-3}, \in = 10^{-4}$ using Quasi Reversibility method

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Table 1.

$ \in $	t = 0		t = 0.05		t = 0.25
	err1	err2	err1	err2	err1	err2
1E-01	5.19E-02	5.11E-03	3.97E-01	1.48E-02	3.90E-01	1.26E-02
1E-02	1.47E-03	1.45E-04	2.90E-02	3.68E-03	9.61E-02	3.12E-02
1E-03	1.89E-05	1.86E-06	1.25E-02	4.63E-04	1.21E-02	3.92E-04
1E-04	1.95E-07	1.92E-08	1.28E-03	4.77E-05	1.24E-03	4.03E-05
1E-05	1.96E-06	1.93E-07	1.29E-04	4.78E-06	1.24E-04	4.04E-06
1E-06	2.05E-07	2.12E-08	1.29E-05	4.78E-07	1.25E-05	4.04E-07

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Table 2.

$ \in $	t = 0		t = 1
	err1	err2	err1	err2
1E-01	4.00E-01	3.04E-02	3.00E-01	1.68E-02
1E-02	4.86E-02	3.33E-02	2.86E-03	1.84E-04
1E-03	4.61E-03	3.25E-04	2.61E-04	1.80E-05
1E-04	5.19E-04	2.43E-05	2.19E-05	1.34E-06
1E-05	6.04E-05	6.88E-07	2.04E-04	3.81E-07
1E-06	7.49E-06	8.43E-08	2.49E-05	4.66E-08

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