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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:
  H. Brezis, Analyse Fonctionelle Masson, Paris, 1983. 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.  G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well posed problems, Elect. J. Diff. Eqns. , (1994), approx. 9 pp. 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.  L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442. doi: 10.1155/S0161171203301486.  X. L. Feng, L. 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.  M. Ginoa, S. 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. J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations Yale University Press, New Haven, CT, 1923.  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. 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. R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications Dunod, Paris, 1967. 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.  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.  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.  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.  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. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. C. Ren, X. 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.  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.  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.  S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, (1993). 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. D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1–10. J. G. Wang, T. 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.  show all references

##### References:
  H. Brezis, Analyse Fonctionelle Masson, Paris, 1983. 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.  G. W. Clark and S. F. Oppenheimer, Quasireversibility methods for non-well posed problems, Elect. J. Diff. Eqns. , (1994), approx. 9 pp. 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.  L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 54 (2003), 3413-3442. doi: 10.1155/S0161171203301486.  X. L. Feng, L. 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.  M. Ginoa, S. 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. J. Hadamard, Lectures on the Cauchy Problem in Linear Differential Equations Yale University Press, New Haven, CT, 1923.  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. 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. R. Lattés and J. -L. Lions, Méthode de Quasi-réversibilité et Applications Dunod, Paris, 1967. 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.  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.  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.  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.  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. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. C. Ren, X. 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.  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.  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.  S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, (1993). 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. D. D. Trong and N. H. Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron. J. Diff. Eqns., 2006 (2006), 1–10. J. G. Wang, T. 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.   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 Reconstruction results at t = 0 from noisy measurement data: 2D drawing using QBV Method 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 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 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 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 Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $\in =10^{-1}, \in =10^{-2}$ using Quasi Reversibility method Reconstruction results at t = 0 from noisy measurement data at $t=1$ with $\in = 10^{-3}, \in = 10^{-4}$ using Quasi Reversibility method
 $\in$ 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
 $\in$ 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
 $\in$ 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
 $\in$ 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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