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March  2017, 6(1): 111-134. doi: 10.3934/eect.2017007

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.

Received  February 2016 Revised  September 2016 Published  December 2016

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.

Citation: 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
References:
[1]

H. Brezis, Analyse Fonctionelle Masson, Paris, 1983.  Google Scholar

[2]

J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[3]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well posed problems, Elect. J. Diff. Eqns. , (1994), approx. 9 pp.  Google Scholar

[4]

M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl., 301 (2005), 419-426.  doi: 10.1016/j.jmaa.2004.08.001.  Google Scholar

[5]

L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442.  doi: 10.1155/S0161171203301486.  Google Scholar

[6]

X. L. FengL. Elden and C. L. Fu, Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region, J. Math. Comp. Simulation, 79 (2008), 177-188.  doi: 10.1016/j.matcom.2007.11.005.  Google Scholar

[7]

M. GinoaS. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[8]

J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations Yale University Press, New Haven, CT, 1923. Google Scholar

[9]

Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Res., 34 (1998), 10271033.  doi: 10.1029/98WR00214.  Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differnetial Equations, North -Holland Mathematics Studies, vol. 204, Elsevier Science B. V, Amsterdam, 2006.  Google Scholar

[11]

R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications Dunod, Paris, 1967.  Google Scholar

[12]

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

[13]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.  doi: 10.2478/s13540-012-0010-7.  Google Scholar

[14]

Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calc. Appl. Anal., 14 (2011), 110-124.  doi: 10.2478/s13540-011-0008-6.  Google Scholar

[15]

R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.  Google Scholar

[16]

R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[18]

C. RenX. Xu and S. Lu, Regularization by projection for a backward problem of the timefractional diffusion equation, J. Inverse Ill-Posed Probl., 22 (2014), 121-139.  doi: 10.1515/jip-2011-0021.  Google Scholar

[19]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[20]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusionwave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[21]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, (1993).  Google Scholar

[22]

R. E. Showalter, Quasi-reversibility of first and second order parabolic evolution equations, in Improperly posed boundary value problems (Conf., Univ. New Mexico, Albuquerque, N. M., 1974), Res. Notes in Math., Pitman, London, 1 (1975), 76-84.  Google Scholar

[23]

D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1–10.  Google Scholar

[24]

J. G. WangT. Wei and B. Y. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Model., 37 (2013), 8518-8532.  doi: 10.1016/j.apm.2013.03.071.  Google Scholar

show all references

References:
[1]

H. Brezis, Analyse Fonctionelle Masson, Paris, 1983.  Google Scholar

[2]

J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.  doi: 10.1016/0370-1573(90)90099-N.  Google Scholar

[3]

G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well posed problems, Elect. J. Diff. Eqns. , (1994), approx. 9 pp.  Google Scholar

[4]

M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems, J. Math. Anal. Appl., 301 (2005), 419-426.  doi: 10.1016/j.jmaa.2004.08.001.  Google Scholar

[5]

L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442.  doi: 10.1155/S0161171203301486.  Google Scholar

[6]

X. L. FengL. Elden and C. L. Fu, Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region, J. Math. Comp. Simulation, 79 (2008), 177-188.  doi: 10.1016/j.matcom.2007.11.005.  Google Scholar

[7]

M. GinoaS. Cerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials, Phys. A, 191 (1992), 449-453.  doi: 10.1016/0378-4371(92)90566-9.  Google Scholar

[8]

J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations Yale University Press, New Haven, CT, 1923. Google Scholar

[9]

Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Res., 34 (1998), 10271033.  doi: 10.1029/98WR00214.  Google Scholar

[10]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differnetial Equations, North -Holland Mathematics Studies, vol. 204, Elsevier Science B. V, Amsterdam, 2006.  Google Scholar

[11]

R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications Dunod, Paris, 1967.  Google Scholar

[12]

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

[13]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal., 15 (2012), 141-160.  doi: 10.2478/s13540-012-0010-7.  Google Scholar

[14]

Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calc. Appl. Anal., 14 (2011), 110-124.  doi: 10.2478/s13540-011-0008-6.  Google Scholar

[15]

R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107-125.  doi: 10.1016/S0378-4371(99)00503-8.  Google Scholar

[16]

R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133 (1986), 425-430.  doi: 10.1002/pssb.2221330150.  Google Scholar

[17]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[18]

C. RenX. Xu and S. Lu, Regularization by projection for a backward problem of the timefractional diffusion equation, J. Inverse Ill-Posed Probl., 22 (2014), 121-139.  doi: 10.1515/jip-2011-0021.  Google Scholar

[19]

H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017.  Google Scholar

[20]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusionwave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar

[21]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, (1993).  Google Scholar

[22]

R. E. Showalter, Quasi-reversibility of first and second order parabolic evolution equations, in Improperly posed boundary value problems (Conf., Univ. New Mexico, Albuquerque, N. M., 1974), Res. Notes in Math., Pitman, London, 1 (1975), 76-84.  Google Scholar

[23]

D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1–10.  Google Scholar

[24]

J. G. WangT. Wei and B. Y. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation, Appl. Math. Model., 37 (2013), 8518-8532.  doi: 10.1016/j.apm.2013.03.071.  Google Scholar

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
Figure 2.  Reconstruction results at t = 0 from noisy measurement data: 2D drawing using QBV Method
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
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
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
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
Figure 7.  The exact solution in Example 2 at t = 1.
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
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
Table 1.   
$ \in $t = 0t = 0.05t = 0.25
err1 err2 err1err2 err1 err2
1E-015.19E-025.11E-033.97E-011.48E-023.90E-011.26E-02
1E-021.47E-031.45E-042.90E-023.68E-039.61E-023.12E-02
1E-031.89E-051.86E-061.25E-024.63E-041.21E-023.92E-04
1E-041.95E-071.92E-081.28E-034.77E-051.24E-034.03E-05
1E-051.96E-061.93E-071.29E-044.78E-061.24E-044.04E-06
1E-062.05E-072.12E-081.29E-054.78E-071.25E-054.04E-07
$ \in $t = 0t = 0.05t = 0.25
err1 err2 err1err2 err1 err2
1E-015.19E-025.11E-033.97E-011.48E-023.90E-011.26E-02
1E-021.47E-031.45E-042.90E-023.68E-039.61E-023.12E-02
1E-031.89E-051.86E-061.25E-024.63E-041.21E-023.92E-04
1E-041.95E-071.92E-081.28E-034.77E-051.24E-034.03E-05
1E-051.96E-061.93E-071.29E-044.78E-061.24E-044.04E-06
1E-062.05E-072.12E-081.29E-054.78E-071.25E-054.04E-07
Table 2.   
$ \in $t = 0t = 1
err1 err2 err1 err2
1E-014.00E-013.04E-023.00E-011.68E-02
1E-024.86E-023.33E-022.86E-031.84E-04
1E-034.61E-033.25E-042.61E-041.80E-05
1E-045.19E-042.43E-052.19E-051.34E-06
1E-056.04E-056.88E-072.04E-043.81E-07
1E-067.49E-068.43E-082.49E-054.66E-08
$ \in $t = 0t = 1
err1 err2 err1 err2
1E-014.00E-013.04E-023.00E-011.68E-02
1E-024.86E-023.33E-022.86E-031.84E-04
1E-034.61E-033.25E-042.61E-041.80E-05
1E-045.19E-042.43E-052.19E-051.34E-06
1E-056.04E-056.88E-072.04E-043.81E-07
1E-067.49E-068.43E-082.49E-054.66E-08
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